A MATHEMATICAL AND STATISTICAL ANALYSIS OF THE -3/2 POWER RULE OF SELF-THINNING IN EVEN-AGED PLANT POPULATIONS A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Donald Ernest Weller June 1985 i i Dedicated to my grandparents Della Florence (Miller) Weller and Everett Earl Weller iii ACKNOWLEDGMENTS The successful completion of a doctoral program requires the assistance of many individuals. In my case, particular thanks are due to Bob Gardner, Hank Shugart, Jr., Don DeAngelis, and Bob o?Neill for their guidance in this and other projects undertaken during my graduate study at Oak Ridge National Laboratory. Committee members Tom Hallam, Dewey Bunting, and Cliff Amundsen also provided invaluable assistance in designing my graduate program and preparing this dissertation. I thank the Graduate Program in Ecology for providing the basic matrix for my graduate education. My fellow graduate students, who are to numerous to be listed here, provided an intellectually stimulating atmosphere that was a essential component of the graduate school experience. Dewey Bunting and Cornell Gilmore helped me to avoid numerous bureaucratic pitfalls and facilitated my graduate program in many other ways. Ray Millemann provided similar assistance in coordinating the interactions between the University of Tennessee and Oak Ridge National Laboratory. Carole Hom, Yetta Jager, and Tom Smith helped me to meet important deadlines after my departure from Tennessee. iv The staff and management of Oak Ridge National Laboratory, particularly the Environmental Sciences Division, maintained an outstanding working environment that included the routine provision of facilities and services unavailable to most graduate students. The Word Processing Center of the Environmental Sciences Division prepared the final copy of the dissertation. I thank my family for encouraging my education, particularly my parents Lowell and Eloise Weller and my grandparents Everett and Della Weller. My wife Debbie deserves special thanks for her love, friendship, encouragement, and continued patience during the difficult stages of my graduate career. I thank the scientists whose work provided the background for this study. They developed the ideas and data vital to a review of this kind, and many sent additional information to extend and clarify published reports. Since this group is too numerous to be listed here, the "Literature Cited" section should be considered an extended list of acknowledgements. Without the studies cited there, my project would have had neither motivation nor basis. This research was supported by the National Science Foundation?s Ecosystems Studies Program under Interagency Agreement BSR-8315185 with the U.S. Department of Energy, under contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. v ABSTRACT The self-thinning rule for even-aged plant populations (also called the -3/2 power law or Yoda?s law) is reviewed. This widely accepted but poorly understood generalization predicts that, through time, growth and mortality in a crowded population trace a straight thinning line of slope -3/2 in a log-log plot of average plant weight versus plant density. The evidence for this rule is examined, then reanalyzed to objectively evaluate the strength of support for the rule. Mathematical models are constructed to produce testable predictions about causal factors. Major problems in the evidence for the thinning rule include inattention to contradictory data, lack of hypothesis testing, inappropriate curve-fitting techniques, and the use of an invalid data transformation. When these problems are corrected, many data sets thought to corroborate the rule do not demonstrate any size-density relationship. Also, the variations among thinning slopes and intercepts are much greater than currently accepted, many slopes disagree quantitatively with the thinning rule, and thinning slope and intercept differ among plant groups. The models predict that thinning line slope is determined by the allometry between area occupied and plant weight, while the vi intercept is also related to the density of biomass per unit of space occupied and the partitioning of resources among competing individuals. Statistical tests confirm that thinning slope is correlated with several measures of plant allometry and that variations in thinning slope among plant groups reflect allometric differences. The ultimate thinning line, which describes the overall size-density relationship among populations of many species, is a trivial geometric consequence of packing objects onto a surface. This cause differs from the factors positioning the self-thinning lines of individual populations, so the existence of an overall relationship is not relevant to the thinning rule. The evidence does not support acceptance of the self-thinning rule as a quantitative biological law. The slopes and intercepts of size-density relationships are variable, and the slopes can be explained by simple geometric arguments. vii TABLE OF CONTENTS CHAPTER INTRODUCTION 1. LITERATURE REVIEW ??????????? Statement of the Self-thinning Rule ????. Evidence for the Self-thinning Rule ?????? Effects of Environmental Factors on Self-thinning Lines . . . ? ? ? . . . . . . . . . . . ? . Explanations for the Self-thinning Rule ??? Interpretation of the Self-thinning Constant K . . . 2. SIMPLE MODELS OF SELF-THINNING .?.?????? . . . 3. 4. Introduction ???????.?????.??? Model Formulation ????????????? Basic Model for the Spatial Constraint ??? Enhanced Model with Additional Constraints ??????. Model Analysis and Results ? ? ????.?? Basic Model ?? Enhanced Model ???? Discussion ? ? . ? . . . . . . Basic Model ? Enhanced Model SIMULATION MODEL OF SELF-THINNING Introduction ? ? ? ? ??? Model Formulation Model Analysis ?????? Results Discussion ?? SOME PROBLEMS IN TESTING THE SELF-THINNING RULE . . . . Introduction ? . ? ? ? ? ? ? ? ? ? ? . . . Selecting Test Data ????????.????? Editing Data Sets ? ? ? ? ? ? ? ? ? ? ? ? ? ? Fitting the Self-thinning Line ??.?.???? Choosing the Best Mathematical Representation ?.?. Testing Agreement with the Self-thinning Rule Discussion ??.??????.??.???.?. PAGE 6 6 11 13 15 18 20 20 21 21 28 29 29 37 38 38 42 47 47 47 53 57 70 75 75 75 77 83 87 96 99 viii CHAPTER SELF-THINNING RULE 5. NEW TESTS OF THE Introduction ? Methods ? ? ? Results ??? Discussion ?? . . . . . . . . 6. SELF-THINNING AND PLANT ALLOMETRY ????? Introduction ?????????????????? . . . Expected Relationships between Thinning Slope and Plant Allometry ?? ~ ? ? ? ? ? ? ??? Methods . ? ? ? ? ? ? ? ? ? ? ? ? . ? ? ? . ? ? Results ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .?? Discussion ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .? 7. THE OVERALL SIZE-DENSITY RELATIONSHIP Introduction ?? o ??? o o o o o o ????? Models of the Overall Size-density Relationship Heuristic Model ???????? Improved Model Discussion ?????? 8. CONCLUSION ? LITERATURE CITED APPENDIXES ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? APPENDIX A. EXPERIMENTAL AND FIELD DATA APPENDIX B. FORESTRY YIELD TABLE DATA ? VITA ? ? ? ? ? ? . . . . . . PAGE 102 102 103 109 120 128 128 128 133 134 151 158 158 159 159 161 164 179 184 198 199 218 240 ix LIST OF TABLES TABLE PAGE 3.1. Competition Algorithms in the Simulation Model ? . ? . . . 51 3.2. Simulation Model Parameters and Their Reference Values . ? 56 3.3. Self-thinning Lines for the Simulation Experiments ? ? 59 3.4. Thinning Line Statistics for the Simulation Experiments 60 4.1. Three Examples of the Sensitivity of the Fitted Self- thinning Line to the Points Chosen for Analysis 81 5.1. Statistical Distributions of Thinning Line Slope and Intercept ? ? ? ? ? ? ? ? ? ? ? . ? ? ? ? ? ? ? ? 110 5.2. Comparisons of Thinning Line Slope and Intercept Among Plant Groups ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . 113 5.3. Spearman Correlations of Shade Tolerance with Thinning Line Slope and Intercept in the Forestry Yield Data 115 5.4. Thinning Line Slopes and Intercepts of Species for which Several Thinning Lines Were Fit from Experimental or Field Data ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 116 5.5. 6.1. 6.2. 6.3. Statistics for Thinning Line Slope and Intercept of Species for which Several Thinning Lines Were Fit from Forestry Yield Data ????.????? Statistical Distributions of Allometric Powers for Experimental and Field Data ??.????? Statistical Distributions of Allometric Powers for Forestry Yield Table Data ??????????? Correlations and Regressions Relating Transformed Thinning Slope to Allometric Powers .???. . . . . . 6.4. Tests for Differences in Allometric Powers Among 121 136 137 140 Groups in the Experimental and Field Data .????.?? 144 X TABLE PAGE 6.5. Tests for Differences in Allometric Powers Among Groups in the Forestry Yield Table Data . . ? ? ? . . ? ? ? ? . 147 6.6. Spearman Correlations of Shade Tolerance with Allometric Powers in the Forestry Yield Table Data ? . ? ? ? 149 7. 1. Parameter Ranges for Monte Carlo Analysis of the Improved Model ?????????.?????? 7.2. Ranges of Variation of Variables Derived in the Monte Carlo Analysis of the Improved Model ? 7.3. Statistical Distributions of K and log K ? A.l. Sources of Experimental and Field Data ?? . . . 165 166 173 200 A.2. Self-thinning Lines Fit to Experimental and Field Data ? ? 203 A.3. Ranges of Log B and Log N Used to Fit Self-thinning Lines to Experimental and Field Data ? ? ? ? ? ? ? ? 207 A.4. Fitted Allometric Relationships for Experimental and Field Data. . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . 211 A.5. Self-thinning Lines from Experimental and Field Data Cited by Previous Authors as Support for the Self-thinning Rule ? ? ? ? ? ? ? ? 216 B. 1. Sources of Forestry Yield Table Data ?????? . . . . . 219 B.2. Self-thinning Lines Fit to Forestry Yield Table Data ? 221 B.3. Fitted Allometric Relationships for Forestry Yield Table Data ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . ? ? . ? . ? 230 B.4. Self-thinning Lines from Forestry Yield Table Data Cited by Previous Authors in Support of the Self-thinning Rule ? ? . ? . ? ? ? ? ? ? ? ? ? . ? ? . . . ? ? . ? . ? 239 xi LIST OF FIGURES FIGURE PAGE 1.1. Four possible growth stages for an idealized even-aged plant population ? ? ? ? ? ? ? ? . ? ? 9 2.1. Power functions of the crowding index 27 2.2. Vertical distance between the zero isocline of biomass growth and the self-thinning line ? ? ? ? ? ? 35 2.3. Basic model fitted to data from a Pinus strobus plantation ?????????????.? . . . . . 2.4. Analysis of a model with three constraints on 2.5. population growth ???????? Effect of a reduction in illumination on population trajectories in a two-constraint model ????? . . . . . . . 36 39 45 3. 1. Typical dynamic behavior of the simulation model ? . 58 3.2. Variations in the self-thinning trajectory due to stochastic factors in the simulation model ??? 3.3. Effect of the allometric power, p, on the self-thinning 1 i ne ? ? ? ? ? ? ? ? ? ? ? ? ? 3.4. Effect of the density of biomass in occupied space, d, 62 63 on the self-thinning line ? ? ? . ? ? ? ? ? ? ? 64 3.5. Effect of the competition algorithm on the self-thinning 1 i ne ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 6 6 3.6. Simulations for three parameters that did not affect the self-thinning line ? ? ? ? ? ? ? ? ? ? ? ? ? . 68 3.7. Dynamics of the weight distributions in four simulations with different competition algorithms ? ? ? ? . ? ? 73 xii FIGURE PAGE 4. 1. Two forestry yield tables showing variation in thinning line parameters with site index . ? ? ? ? ? ? . ? ? ? . 78 4.2. Three examples of the sensitivity of self-thinning line parameters to the points chosen for analysis ? ? ? ? ? . 80 4.3. Example of a data set with two regions of linear behavior ? ? ? ?????????.???? 82 4.4. Thinning lines fit to one data set by four methods ? . 85 4.5. Example of a spurious reported self-thinning line 91 4.6. Example of potential bias in data editing in the log w-log N plane ? ? ? ? ? ? ? . ? ? . . 92 4.7. Example of deceptive straightening of curves in the log w-log N plane ? ? ? ? ? ? ? . ? ? ? . ? ? ? ? 94 4.8. Potential misinterpretation of experimental results due to deceptive effects of the log w-log N plot ? 95 5. 1. Histograms for the slopes and intercepts of fitted thinning lines ?????????????.??. 112 6. 1. Histograms of allometric powers of the experimental and field data ? ? ? ? ? ? ? ? ? ? ? . . ? ? . ? ? . ? ? ? ? 138 6.2. Histograms of allometric powers of the forestry yield table data ?.???.?.??.......?... . . 139 6.3. Observed relationships between transformed thinning slope and allometric powers in the experimental and field data 141 6.4. Observed relationships between transformed thinning slope and allometric powers in the forestry yield table data ? 143 7. 1. Monte Carlo analysis of the improved model for the overall size-density relationship ? ? . . ? ? ? ? . ? . 167 xiii FIGURE PAGE 7.2. Previous analysis of the overall size-density relationship among thinning lines . ? ? ? ? ? ? ? ? ? . ? ? ? ? . ? ? 170 7.3. Gorham?s analysis of the overall size-density relationship among stands of different species ? ? ? ? ? ? ? 171 7.4. Histograms of log K . . . . . . . . 174 SYMBOL ANOVA BSLA CI cv DBH EFD %EV FYD GMR ln log, loglO PCA RGR RMR RHS SD SE xiv LIST OF ABBREVIATIONS AND MATHEMATICAL SYMBOLS ABBREVIATIONS MEANING analysis of variance basal area of tree bole at breast height confidence interval coefficient of variation diameter of tree bole at breast height experimental and field data percentage of the summed variance of two variables explained by bivariate PCA forestry yield table data geometric mean regression natural logarithm common logarithm principal component analysis relative growth rate relative mortality rate right hand side of an equation or inequality standard deviation standard error SYMBOL VOI ZOI XV MEANING volume of influence; space filled zone of influence; area covered (bar) indicates average value, e.g. w, a sign of proportionality infinite; without limit indicates a statistical estimate of a parameter, e.g. S, & xvi MATHEMATICAL SYMBOLS SYMBOL CHAPTER MEANING a 1-3,6-7 growing surface area occupied by a plant a1oss 3 area of a plant ZOI lost to a competitor A 3 total area covered by a population Aplot 3 size (area) of a simulated plot b 1,3 constant in a power function relating maintenance cost to weight; cost = b wQ B all total population biomass; stand biomass Bmax 2 maximum biomass that a population can attain; carrying capacity for biomass c 2,3,7 constant in a power function relating ZOI radius to weight c1 2,3,7 constant defined by c1 = TI c2 c2 2 constant correcting for systematic differences between the a-w and a-w power relationships c3 2 constant related to the allowable overlap between plant ZOis C(N,B) 2 function for crowding at given biomass and density Cov(X,Y)4 d f 3 3 2 covariance of variables X and Y density of biomass per unit volume occupied distance between the ith and jth plants in a population a constant in the crowding function C(N,B) SYMBOL CHAPTER F 5~6 go 2 gl 1~3 G(N~B) 2 Gr(N~B) 2 H 5~6 h 2-7 K all m 2 M{N,B) 2 Mr(N~B) 2 N all n 1-7 nt 3 no 3 p 1-7 Pest 6 xvii MEANING the variance ratio computed in ANOVA maximum relative growth rate of biomass maximum assimilation rate per unit of ground area covered function for the growth rate of population biomass multiplier function reducing RGRs with increasing crowding test statistic from Kruskal-Wallis analysis plant height constant in the self-thinning rule equation mortality constant function for population mortality rate multiplier function producing higher relative mortality with increasing crowding plant density in individuals per unit area size of a sample for statistical analysis number of plants remaining in a simulation at time t initial number of plants in a simulation power relating ZOI radius to plant weight; R = c wP power relating ZOI radius plant weight~ estimated by mathematical transformation of a fitted thinning slope; Pest= 0.5 I (1-8) SYMBOL CHAPTER p q r R RGR RGRs RGRw 1-7 3 3-7 3-7 3,6 1-7 1-3 1-3 1-3 RGRrni n 3 RMR 1-3 sw0 3 Sxx' Syy, 4 Sxy t 2,3 Var( X) 4 xviii MEANING statistical significance level; probability of a Type I statistical error power relating maintenance cost to weight; cost = b wq correlation coefficient coefficient of determination Spearman rank correlation coefficient ZOI radius relative growth rate relative growth rate of total biomass relative growth rate of average weight minimum individual RGR allowing survival relative mortality rate standard deviation of weights at time 0 sums of squares and cross products for X and Y time variance of variable X v 1-3,6-7 volume of an individual plant VOI w all weight of an individual plant w all average plant weight for a population SYMBOL CHAPTER wo 3 Wmax 2 X, y 4 a. 1-6 a.3.35 3 t3 all St 2 Syx 4 y all 6. log K 2 !J.t 3 ( !J.t )m 3 e: 3 n 3 l; 7 xix MEANING initial plant weight at time 0 maximum individual weight general symbols for two variables involved in a bivariate relationship common log of K (a.= log K); intercept at log N = 0 of the self-thinning line intercept of the self-thinning line at log N = 3.35 power in the B-N thinning equation B = K Ni3; slope of the log B-log N self-thinning line instantaneous slope of the self-thinning trajectory at time t slope of a linear function giving Y in terms of X power in the w-N thinning equation w = K NY; slope of the log w-log N self-thinning line difference between the log N = 0 intercepts of the straight lines in the log B-log N plane representing the asymptotic population trajectory and the zero isocline of biomass increase simulation time step simulation time step over which RGR?s are averaged to identify plants with untenably low growth rates expansion factor to correct for edge effects in the simulation model sample angle in edge correction equation parameter of the lognormal distribution; if w is lognorma 1, t; is the mean of ln w XX SYMBOL CHAPTER MEANING ~1 - 65 2 parameters controlling the suddenness of onset of model constraints 4 ratio of the residual variances in Y and X around a K 7 p 7 CJ 7 T 3,7 sw 6 Bw allometrjc power relating density to weight; d ex: w ~dw (density of biomass per unit of volume occupied) allometric power relating DBH to weight; DBH a: w hB allometric power relating height to DBH; h ex: DBHhw SYMBOL CHAPTER 2 w 2 XX i MEANING slope of a straight line in the log B-log N plane representing points where d log B I d log N = ~ ratio of mortality and growth constants; w = m I go INTRODUCTION The title of a recent review article 11 Ecology's law in search of a theory 11 (Hutchings 1983) concisely summarizes the status of a hypothesis variously referred to as the self-thinning rule, the -3/2 power law, or Yoda's law. This rule states that as growth and mortality proceed in a crowded even-aged plant population, average weight (w) and the number of plants per unit area (N) are related by a simple power equation w = K NY, where y = -3/2 and K is a constant. Widespread acceptance of this rule by plant ecologists is based on many observations of this power relationship in plant populations ranging from mosses to trees, but the reasons for the rule's apparent generality are not well understood (Hutchings and Budd 198la, Westoby 1981, White 1981, Hutchings 1983). The theoretical importance of the self-thinning rule is evidenced by the published statements of plant ecologists. White (1981) called it one of the best documented generalizations of plant demography, and further states that its empirical generality is now beyond question. Westoby (1981) considers it the most general principle of plant demography and suggests that it be elevated beyond the status of an empirical generalization to take a .. central place in the concepts of population dynamics. 11 Hutchings and Budd ( 198la) emphasized the uniqueness of its precise mathematical formulation to a science where most general statements can be stated in only vaguer, qualitative terms. To many ecologists, the rule is 2 sufficiently well verified to be considered a scientific law (Yoda et al. 1963, Dirzo and Harper 1980, Lonsdale and Watkinson 1982, Malmberg and Smith 1982, Hutchings 1983). Harper (quotea in Hutchings 1983) called it "the only generalization worthy of the name of a law in plant ecology 11 and Mcintosh (1980) agreed that, if substantiated, a self-thinning law could well be the first basic law demonstrated for ecology. As such, it would help to fill the need for verified laws in a discipline that has been hindered by a lack of laws and other regularities from which to develop the body of comprehensive theory that is the hallmark of a mature science (Johnson 1977, Mcintosh 1980). There have been many proposed extensions and applications of the self-thinning rule. Although it was originally observed in monocultures, evidence now suggests that aggregate measurements of even-aged two-species mixtures (White and Harper 1970, Bazzaz and Harper 1976, Malmberg and Smith 1982) and many-species mixtures {White 1980, Westoby and Howell 1981) also conform. Some authors have even attempted to apply the thinning rule to animal populations (Furnas 1981, Wethey 1983). The rule can be a used to compare the site qualities or histories of plant populations growing at different sites, since relative positions along the curve w = K N- 312 give a ranking of the fertilities experienced by equal-aged populations (Yoda et al. 1963), or of population ages if fertility is constant among sites (Barkham 1978). The existence of the -3/2 power relationship among a set of population measurements has been interpreted as evidence that natural self-thinning is 3 occurring, canopies are closed, growth and mortality are ongoing, and competition is the cause of mortality (Barkham 1978), while the absence of the -3/2 relationship has been cited as evidence that density independent factors are the important causes of mortality (Schlesinger 1978). The rule can also be a useful management tool in forestry (see Yoda et al. 1963, Drew and Flewelling 1977, and Japanese language papers cited therein), or in other applications requiring predictions of the limits of biomass production for a given species at any density (Hutchings 1983). In view of the popularity, theoretical importance, and applicability of the self-thinning rule, the troublesome lack of a verified explanatory theory continues to motivate further research. The original objective of the present study was to examine the causes of the self-thinning rule by analyzing a detailed simulation model of a generalized plant population and extracting testable predictions about the effects of different biological parameters on the constants y and K of the power rule equation. The central question was how the processes of growth and mortality operate to eliminate large among-species variations in size, shape, physiological capability, strategy, and other important factors so that all species obey the same power rule. The model developed for this purpose did not predict such constancy, rather it suggested that the power y should vary from the idealized value of -3/2, with its exact value determined by the relationship between ground area covered and plant weight in a given stand. This hypothesis is 4 neither new (see Miyanishi et. al 1979) nor well supported in previous studies (Westoby 1976, Mohler et al. 1978, White 1981). Among the ideas considered after reaching this conclusion was the thought that the previous analyses were somehow wrong. When the evidence behind the self-thinning rule was carefully examined, some troubling problems were discovered. They motivated a new analysis of the data supporting the thinning rule and some new tests of the simple hypothesis suggested by the simulation model. Together, the mathematical models and data analyses evolved into six lines of investigation presented in Chapters 2 through 7 of this report. Chapter 2. Simple, spatially-averaged models of growth and mortality are developed to formalize in a dynamic model the hypothesis that the power of the self-thinning equation is determined by a relationship between ground area covered and plant weight, and to examine the interaction between this effect and constraints imposed by a carrying capacity and a maximum individual size. Chapter 3. A detailed, spatially explicit computer simulation model is analyzed to consider the effects of additional population parameters on the power rule relationship, and to verify that the conclusions of the simpler models are not artifacts of spatial averaging. Chapter 4. Some difficulties in the selection, analysis, and interpretation of self-thinning data are discussed along with the implications for tne self-thinning rule. Remedies for some problems are suggested. 5 Chapter 5. Biomass and density data for 488 self-thinning relationships are statistically analyzed to evaluate support for the self-thinning rule and to estimate the variability of the power equation parameters K and y. Variations in these parameters among plant groups are considered. Chapter 6. The biomass and density data from Chapter 5 are combined with plant shape measurements from the same stands and used to test some predictions of the hypothesis that the power of the self-thinning equation is determined by a relationship between ground area covered and plant weight. Chapter 7. Some hypotheses are presented to explain the overall relationship between size and density. (This relationship emerges when measurements of many crowded plant stands ranging from small herbs to trees are combined to estimate a single power equation relating stand size and stand density across the entire plant kingdom--Gorham 1979, White 1980). The relevance of this overall relationship to the self-thinning rule for individual plant populations is considered. These efforts are united by an overall goal of developing and testing a unified theoretical framework for understanding observed size-density relationships and evaluating their scientific importance. 6 CHAPTER LITERATURE REVIEW Statement .Q_f_ the Self-thinning Rule The self-thinning rule describes a relationship between size and density in even-aged plant populations that are crowded but actively growing. In the absence of competition from other populations and of significant density-independent stresses (drought, fire, etc.), mortality or 11 thinning 11 is caused by the stresses of competition within the population, hence the term 11 Self-thinning. 11 Yoda et al. (1963) observed a general relationship among successive measurements of average weight and density taken after the start of self-thinning. When the logarithm of average weight is plotted vertically and the logarithm of plant density horizontally, the points form a straight line with a slope near -3/2 represented by the equation log w = y log N + log K where w is average weight (in g), N is plant density (in individuals/m2), y = -3/2, and K is a constant. Antilogarithmic transformation of this relationship gives the power equation ( 1. 1 ) ( 1. 2) Since w = B/N, (where B is total stand biomass or yield in g/m2), these equations relating average weight to density are 7 mathematically equivalent to the following relationships between stand biomass and density: 1 og B = t3 1 og N + 1 og K and B = K NB , with 8 = -l/2. The parameters y and 8 are related by 8 = y + 1. ( 1.3) ( 1 ? 4) Yoda et al. (1963) named this relationship the 11 -3/2 power law of self-thinning .. because of the mathematical form of equation 1.2 and the value of the power y. Any of the above relationships can be referred to as the self-thinning equation, while the linear equations 1.1 and 1.3 give the self-thinning line, or simply the thinning line. y and S are called self-thinning powers or exponents in reference to equations 1.2 and 1.4, but the names self-thinning slope or thinning slope are also common because these parameters are the slopes of the linear equations 1.1 and 1.3. The logarithm of the constant K is often called the self-thinning intercept because it gives the intersection of the thinning line with the vertical line log N = 0. The self-thinning rule does not necessarily apply throughout the history of a plant population. When populations are established at combinations of average weight and density well below the self-thinning line, the initial competitive stresses can be absorbed through plastic changes in shape, growth can proceed with little 8 mortality, and a nearly vertical line is traced in a log-log plot of size versus density. As competition becomes more severe, growth slows and some plants die after exhausting their capacities to adjust to their neighbors? intrusions. The population?s path in log size-log density plot bends toward the self-thinning line, then begins to move along it as growth and mortality progress (Hutchings and Budd 198la, White 1981). The curves traced by populations of differing initial densities converge on the same self-thinning line (Yoda et al. 1963, White 1980, 1981). Movement along the self-thinning line can not continue indefinitely because the population will eventually approach the environmental carrying capacity, which places an upper limit on total stand biomass (Hutchings and Budd 198la, Peet and Christensen 1980, Lonsdale and Watkinson 1983b, Watkinson 1984). Limits on individual growth, such as a genetically or mechanically defined maximum size or the end of the growing season, can also deflect the population from the thinning line (Harper and White 1971, Peet and Christensen 1980). The history of an even-aged population can, then, be divided into up to four stages: (1) a period of initial establishment, rapid growth, and low mortality; (2) a period of adherence to the self-thinning rule; (3) a period when constant biomass is maintained at the carrying capacity; and (4) a period of population degeneration, when growth does not replace the biomass lost through mortality. Figure 1.1 shows the idealized representions of these stages as paths in the log w-log Nand log B-log N planes. (a) 3 (b) 5 4 ,----.., 0) 3 '---" 3 -+- _c 2 0) ,----.., 4 N E 2 ~ Q) 0) 3: '---" Ul Q) 2 Ul 0 4 0) 0 ~ E 0 Q) > m <( S' 0 0) 0 _J S' 0) 0 _J -1 3 2 3 4 2 3 4 Log,0 Density (pI ants/ m 2) Log 10 Density (pI ant s/m 2 ) Figure 1.1. Four possible growth stages for an idealized even-aged plant population. (1) initial rapid growth and low mortality, (2) self-thinning rule, (3) constant biomass at carrying capacity, and (4) terminal degeneration or senescence. In the log w-log N plane, (a) the respective slopes of the last three stages are -3/2, -1, and 0, while the corresponding slopes in the log B- log N plane (b) are -1/2, 0, and +1. In both plots, the line representing the first stage is nearly vertical. 10 The self-thinning rule describes a certain balance between the rates of growth and mortality, which are linked so that the ratio of the relative growth rate to the relative mortality rate is held constant (Harper and White 1971, Westoby and Brown 1980) at the value of the self-thinning slope, that is, = dw 1 (w dt) dN I (N dt) = d log w d 1 og N = y ' where RGRw is the relative growth rate of average weight and RMR is the relative mortality rate (Hozumi 1977). The equivalent relation for the relative growth rate of biomass, RGRB, is RGRB = dB I (B dt) RMR dN I (N dt) = d 1 og B d 1 og N = s . Mortality during self-thinning is concentrated among smaller ( 1 ? 5) ( l ? 6) individuals that have been suppressed, possibly because of smaller seed size, later germination, lower growth rate, or close neighbors (Ross and Harper 1972, Kays and Harper 1974, Ford 1975, Hutchings and Barkham 1976, Bazzaz and Harper 1976, Harper 1977, Mohler et al. 1978, Rabinowitz 1979, Hutchings and Budd 198la, Lonsdale and Watkinson l983a). These deaths permit a net biomass production among the remaining plants so that total population biomass increases (Ford 1975, Hutchings and Barkham 1976, Westoby 1981). The self-thinning line can also be interpreted as a constraint separating possible combinations of biomass and density from impossible ones. Biomass and density combinations below or on the 11 line are possible, while combinations above the line do not occur (Yoda et al. 1963; White 1980, 1981; Westoby and Howell 1981). This means that thinning lines need not be measured by following individual populations through time, rather data for a given population type from several plots of different ages can be used, provided that the plots do not differ in important biotic or environmental factors that would alter the position of the thinning line. This method has been used since the earliest studies (Yoda et al. 1963), and is particularly important for studying long-lived trees (White 1981). Evidence for the Self-thinning Rule The self-thinning rule is an empirical statement generalized from repeated observations of linear relationship between log w-log N with an estimated slope near -3/2. The supporting examples include populations ranging in size from small herbs to large trees collected from experimental or natural conditions or from forestry yield tables. Yoda et al. (1963) presented ten data sets which exhibit such a relationship. White and Harper (1970) added five additional examples, plus some evidence from forestry thinning tables. White (1980) presented 36 more examples and mentioned unpublished data for 80 additional cases. White?s paper has been widely cited by other authors, along with the study of Gorham (1979), as firmly establishing the generality of the -3/2 rule (Hutchings 1979, Dirzo and Harper 1980, Furnas 1981, Westoby 12 and Howell 1982, Lonsdale and Watkinson 1982, 1983a, Watkinson et al. 1983). Tables A.5 and B.5 give references to studies presenting additional corroborative examples. Most of the weight measurements in these studies are of aboveground plant parts only, but some also include roots. This inclusion raises the value of K but does not affect the slope of the thinning line (Watkinson 1980, Westoby and Howell 1981). The limited range of variation of thinning line slopes and intercepts are also considered strong evidence for the thinning rule. White (1980) reported that thinning slopes vary between -1.3 and -1.8 when log w is fitted to log N. This range is considered remarkably invariant (Watkinson et al. 1980, Furnas 1981, Lonsdale and Watkinson 1982, Hutchings 1983); however, several exceptions have been reported (Ernst 1979, O'Neill and DeAngelis 1981, Sprugel 1984). The constant K is usually limited to 3.5 ~log K ~ 4.4 and values outside this range are biologically significant (White 1980, 1981). Grasses can give higher log K values between 4.5 (Kays and Harper 1974) and 6.67 (Lonsdale and Watkinson 1983a), possibly because their erect, linear leaves allow deeper light penetration through the canopy and greater plant biomass per unit volume (Lonsdale and Watkinson 1983a, Watkinson 1984). Some values of log K greater than 4.4 have also been reported for herbaceous dicots (Westoby 1976, Westoby and Howell 1981, Lonsdale and Watkinson l983a). 13 Further support for the self-thinning rule has come from two studies that examined the relationship between size and density among stands of different species (Gorham 1979, White 1980}. This relationship is referred to here as the overall size-density relationship to distinguish it from the self-thinning lines of particular populations. Gorham (1979} reported that measurements of 65 stands of 29 species (including mosses, reeds, herbs, and trees) form a straight line of slope -1.5 in the log w-log N plane, while White (1980} showed that the individual self-thinning lines of 27 different species are closely grouped around a common linear trend of slope -1.5 (see Chapter 7). These results are considered evidence that self-thinning rule applies over a wide range of plant types and growth forms (Gorham 1979, White 1980, 1981, Westoby 1981, Malmberg and Smith 1982, Hutchings 1983). Even some types of plant populations that do not trace straight trajectories in the log w-log N plane, such as the shoots of clonal perennial species, still seem to be constrained below the ultimate thinning line described by Gorham's study (Hutchings 1979). Effects of Environmental Factors~ Self-thinning Lines Ecologists have examined the effects of the availability of essential resources, such as light and mineral nutrients, on self-thinning. Plants grown at low levels of illumination thin faster (Harper 1977) and reach maximum biomass levels sooner (Hutchings and Budd l98lb) than populations grown with higher illumination. Decreased illumination also lowers the intercept of 14 the self-thinning line (White 1981, Hutchings and Budd 198la, 198lb, Westoby and Howell 1981), possibly due to a decrease in the density of plant matter per unit of occupied space (Lonsdale and Watkinson 1982, 1983a) or to more rapid mortality among shorter plants (Hutchings 1983). The thinning slope is not affected by mild reductions in illumination, but changes from the typical value of -3/2 to -1 under severely lowered light treatments have been observed (White and Harper 1970, Kays and Harper 1974, Harper 1977, Furnas 1981). However, other experiments with equally severe light reductions report no change in the thinning slope (Westoby and Howell 1981, Hutchings and Budd 198lb). Possible reasons for these ambiguous results are considered by Westoby and Howell (1982) and Lonsdale and Watkinson {1982). Since the level of illumination affects the position of the thinning line, light has been implicated as the limiting factor whose availability controls the rate of self-thinning (Kays and Harper 1974, Harper 1977, Westoby and Howell 1982, Lonsdale and Watkinson 1982). High levels of mineral nutrients increase growth and mortality rates and the rate at which populations approach and follow the self-thinning line. The slope and position of that line are insensitive to difference in fertility (Yoda et al. 1963, White and Harper 1970, Harper 1977, White 1981, Westoby 1981, Hutchings and Budd 198la}; however, there is some evidence against this established view. White (1981) mentioned unpublished data which suggests that fertilizer treatments in forest stands may systematically alter log K, while Hara {1984) showed that forestry 15 yield tables can indicate different thinning lines for stands grown on sites of different quality. Furnas (1981) saw an effect of soil fertility on thinning line position in his own experiments and in those of Yoda et al., which have interpreted as lacking a fertility effect (Yoda et al. 1963, White and Harper 1970). Explanations for the Self-thinning Rule Yoda et al. (1963) derived a simple, geometric explanation of the self-thinning from two assumptions: (1) plants of a given species are always geometrically similar regardless of habitat, size, or age; and (2) mortality occurs only when the total coverage of a plant stand exceeds the available area then acts to maintain 100% cover. The first assumption allows the ground area, a, covered by a plant to be be expressed mathematically as a power function of plant weight, a oc w213, while the second assumption implies that the average area covered is inversely proportional to density, that is, a oc 1/N. Combining these two equations and adding a constant of proportionality, K, gives the thinning rule equation w = K N- 312? Starting from the Clark and Evans (1954) equation for nearest neighbor distance, White and Harper (1970) developed an alternative derivation that renders the second assumption unnecessary, but still implicitly requires the first. The assumption that plant shape is invariant is not tenable, so these derivations of the thinning rule are unsatisfactory (White 1981, Furnas 1981). Miyanishi et al. (1979) attempted to reconcile these simple geometric models with the fact of varying plant shapes 16 in their generalized self-thinning law, which states that the power of the thinning equation depends on the proportionality between plant weight and ground area covered. Their hypothesis can be stated mathematically by setting the area covered proportional to w2P, where p can deviate from l/3 to represent changes in shape with increasing size (allometric growth). With this modification, the Yoda logic gives w = K N-l/( 2P) and the thinning slope is y = -l/(2p), which equals -3/2 only if shape is truly invariant (isometric growth, p = 1/3). Westoby (1976) used similar logic to predict that the thinning slope should be -1 in the special case of plants that grow radially, but not in height. The implication of these modified geometric arguments that the self-thinning slope should be dependent on plant allometry has not been supported by experimental tests. Westoby?s (1976) experiment with plants that grow only radially gave a thinning slope near -3/2, not the expected value of -1; however, White (1981) discredited this result, claiming that that the species used really does grow in height. Mohler et al. (1978) use allometric data for two tree species to predict thinning slopes of -2.17 and -1.85, which disagree with the idealized value of -1.5 and with the respective measured slopes of -1.21 and -1.46. In the most extensive review of the allometric theory to date, White (1981) applied an allometric model to available data for trees and predicted thinning slopes between -2.05 and -0.78. His discussion of this result implies that the allometric model is faulty in predicting of thinning slopes outside the range -1.8 to -1.3 that he considers acceptably close to 1 7 -1.5. White concluded that any theory predicting thinning slopes significantly different from -1.5 is not useful or realistic. Several non-allometric explanations of explanations of the self-thinning rule have also been proposed. Westoby (1977) hypothesized that self-thinning is related to leaf area rather than to plant weight, but this theory has been discredited (White 1977, Gorham 1979, Hutchings and Budd 198lb). Mohler et al. (1978) suggested that the value of -3/2 is maintained by a mutual adjustment of plant allometry and stand structure during self-thinning, while Furnas (1981) proposed that the -3/2 value is determined by the fact that limiting resources for plant growth are distributed in a three-dimensional volume. Jones (1982) hypothesized that the -3/2 thinning relationship derives from growth and mortality acting to maintain a constant total plot metabolic rate. Recently, mathematical models have been used to develop even more theories. Pickard (1983) presented three different models deriving the -3/2 value from a mixture of allometric theory and physiological considerations, such as the fraction of photysynthate allocated to biomass increase, the amount of structural and vascular overhead incurred by spatial extension, and the rise in proportionate maintenance costs with increasing weight. Charles-Edwards (1984) combined his basic hypothesis that each plant requires a minimum flux of assimilate to grow and persist with additional assumptions about the mathematical representation of growth and mortality to produce an explanation of self-thinning. 18 Perry (1984) derived the self-thinning curve from a physiological model in which the relationship between leaf area and weight, the decrease in photosynthetic efficiency with crowding, maximum plant size, and age are all important factors that must obey certain mutual constraints if the model is to give thinning slopes and intercepts within the ranges that have been actually observed. In light of the apparent failure of the allometric theory and the paucity of experimental tests of other theories, no satisfactory explanation of self-thinning rule has yet emerged (White 1980, 1981, Westoby 1981, Hutchings and Budd l98la, Hutchings 1983). It is a 11 Crude statement of constraint whose underlying rationale remains elusive 11 (Harper as quoted in Hutchings 1983). Interpretation gi the Self-thinning Constant ~ Plant ecologists are also interested in interpreting the constant K and its observed range of variation. K has been presented as a species constant invariant to changes in all environmental conditions except the level of illumination (Hickman 1979, Hozumi 1980, White 1981, Hutchings 1983). Many authors regard K as a parameter related to plant architecture (Harper 1977, Gorham 1979, Hutchings and Budd 198la, Lonsdale and Watkinson 1983a), but some have proposed that K is insensitive to plant morphology (Westoby 1976, Furnas 1981). White (1981) suggested that K is a rough approximation of the density of biomass in the volume of space occupied by plants and can be considered as a weight to volume conversion, but Lonsdale and Watkinson (1983a) provided evidence 19 against this hypothesis. Lonsdale and Watkinson (1983a) concluded that plant geometry, particularly leaf shape and disposition, do influence thinning intercepts. Harper (1977) speculated the pyramidally shaped trees have higher intercepts than round crowned trees. Westoby and Howell (1981) and Lonsdale and Watkinson (1983a) have hypothesized that shade tolerant plants should have higher thinning intercepts than intolerant plants. Understanding of K is in a similar status as understanding of the -3/2 power: no general theory explaining the variations in K has yet been developed (Hutchings and Budd 198la). 20 CHAPTER 2 SIMPLE MODELS OF SELF-THINNING Introduction Most proposed explanations of the self-thinning rule have been based on simple geometric models of the way plants occupy the growing surface (Yoda et al. 1963, White and Harper 1970, Westoby 1976, Miyanishi et al. 1979, Mohler et al. 1978, White 1981), but these models suffer from two major limitations. Since they are not related to time dynamics, they can not provide an interpretation of the thinning line as a time trajectory (Hozumi 1977, White 1981). Also, they can not explain deviations from the self-thinning rule, such as the alteration of thinning slopes in deep shade (Westoby 1977). Two dynamic models are developed here to remedy these deficiencies. A basic model, in which growth is constrained only by the limited availability of growing space, gives an asymptotic linear trajectory in the log B-log N plane. The slope and intercept of the model self-thinning line are related to the parameters and assumptions of the model to develop biological interpretations for the slope and intercept. The second model incorporates two additional growth constraints: a upper limit on the size of individuals (Harper and White 1971, Peet and Christensen 1980) and a maximum total population biomass or carrying capacity (Peet and Christensen 1980, Hutchings and Budd 198la, Lonsdale and Watkinson 21 1983b, Watkinson 1984}. The general behavior of this model is related to observed phases of population growth (White 1980), and the effects of heavy shading on trajectories in the log B-log N plane are considered. Model Formulation Basic Model for the Spatial Constraint The first step in formulating a dynamic model of self-thinning is selecting a set of state variables to represent the plant population at any timet. Animal populations have been successfully modeled with a single state variable, such as the total number of animals, because animals are relatively uniform in size and simple counts provide rough estimates of total biomass, growth rates, and productivity (Harper 1977}. However, similar-aged plants can vary up to 50,000-fold in size and reproductive output, so measurements of both numbers and biomass are essential for understanding plant populations (White and Harper 1970, Harper 1977, White 1980, Westoby 1981). Average weight has been the measure of population biomass in most self-thinning analyses, but this popular choice entails some serious statistical difficulties that can be avoided if total biomass is used (see Chapter 4). Accordingly, the models developed here employ two state variables, plant density and stand biomass, as a minimum reasonable representation of a plant population. Density, N(t), and biomass, B(t), are measured in individuals per unit area and weight per unit area, respectively. 22 In an even-aged population with no recruitment, plant density and stand biomass change only through growth and mortality. If the population is sufficiently large, these processes can be modeled by two differential equations, dN dt = -M(N,B) (2.1) and dB = ~ G(N,B) , (2.2) where M and G are functions for the rates of mortality and growth when the population state is (N,B). The simplest choices for these functions would apply to a young population of widely spaced plants. In this case plants would not interfere with one another, growth would be approximately exponential, and mortality would be zero (ignoring density-independent causes of mortality). Equations 2.1 and 2.2 would become dN ~ = 0 (2.3) and dB go B ' dt = (2.4) where g0 is the exponential growth constant in units of time-1? 23 The deleterious effects of crowding can be represented in this model by assuming that competition reduces the growth rate and increases the mortality rate. The reduction in growth rate can be modeled by multiplying the exponential growth rate by a function Gr(N,B) < 1 that decreases as either B or N increases, that is, dB dt = go B Gr(N,B) ? (2.5) To specify a similar function for the increase in mortality with crowding, assume that populations undergoing the same level of crowding stress have the same level of per capita mortality, then define M(N,B) = m Mr(N,B) with m constant. The relative mortality rate is, then, dN N dt = -m Mr(N,B) , (2.6) which is constant for any given level of crowding stress Mr. Mr is near zero in a widely spaced population and increases as either B or N increases. Further model development requires some mathematical representation of crowding. A reasonable assumption is that crowding depends on the amount of space actually occupied by the population relative to the total available space. Assume that the 24 area, a, occupied by an individual is related to its weight, w, by a power function a = ( 2. 7) with 0 ~ p ~ 0.5. Plants that grow only upwards are represented by p = 0, while p = 0.5 gives pure radial growth. The special case of isometric growth where shape does not vary with size is given by p = 1/3. The constant c1 is inversely related to the density of biomass per unit of occupied space. This constant also depends on initial plant shape, with initially shorter but fatter plants having higher values than taller, thinner ones. Now assume a similar relationship between the average area occupied and the average plant weight = -2P w ' (2.8) with constant c2 correcting for any systematic differences between the a-w relationship and the a-w relationship for individuals. The total area, A, occupied by N individuals is proportional to the product of N and the average area occupied, = (2.9) The constant c3 > 0 is related to the allowable overlap between neighboring plants. If overlap is extensive, c3 is less than one and the total area occupied is less that the product of the number 25 of plants and the average area occupied. If plants touch but do not overlap, c3 is near one and A approximately equals N a. A shade-intolerant population would have a higher value of c3 than a shade-tolerant one. Since w = B/N, the total area covered can also be expressed as A = Nl-2p 82p (2. 10) Now divide this expression for the total area occupied by the available area, which is simply one because density is already scaled to individuals per unit area. This gives a general crowding index, C(N,B): C(N,B) = (2.11) with f = c3 c2 cl" The constant, f, in this crowding function subsumes several factors, including a weight to volume conversion (the density of biomass in occupied space) and information on initial plant shape from constant c1 of equation 2.7, and a correction from c2 in equation 2.8 for any systematic differences between the i-w relationship and the a-w relationship for individuals. Also, f includes information from constant c3 on how much overlap between the zones of influence of plants is permissible. The crowding index, C(N,B) can now be used to further specify the functions Gr and Mr by defining Mr(N,B) = C(N,B) (2.12) 26 and Gr(N,B) = 1 - C(N,B) ? (2.13) The differential equation model now becomes ~~ = -m N C(N,B) = -m f N1-2P s2P (2.14) and ~~ = g0 B [1 - C(N,B)] = g0 B [1 - f Nl-2p B2P]. (2.15} Two additional parameters can be used to adjust how rapidly the growth and mortality rates respond to changes in crowding. The linear equations 2.12 and 2.13 are special cases of more general power functions of C, and 8 Mr(N,B} = C(N,B} l = 83 - C(N,B) , (2.16} (2.17} where both e1 and 83 are greater than zero (Figure 2. 1}. The use of such power functions in plant growth models is discussed in Barnes (1977). With these modifications, equations 2.14 and 2.15 become dN dt 8 = -m N ( f N1- 2P B2P ) l (2.18) (a) 2,-----------------------------------~ 0 ' ' ' ' ' ' ' : !/ ---=-=::::~----- 0.0 4,: 2/ / ,/ ' ' / // 0.5 ' ' ,/,'-' ~ --- ~ :::~~.;.? .'?~--------------------------0.2 '' ,'//// ,// ,./ ,/ ...... ' ,/,' ,',// /,' ,/ /,// 0.5 1.0 Crowding Index C 1.5 (b) "' Q) u 0 -1 r\ ___ -= == = =:: ~----- '' '', : ', ~ ' .. " \ ' ........ 0.0 ' ' ',',,<::?\ --- ____ -_-_-_-_-:::~:\ \:,::::::------ 0.2 \._\\, o.5 \ \\\ ' ' ' 2\ ' 4\ 0.5 1.0 Crowding Index C Figure 2.1. Power functions of the crowding index. (a) plots the function Mr =eel against C for the indicated values of e1 while {b) plots Gr = 1 - ce3. The powers e1 and 83 adjust how suddenly each function changes near the critical value C = 1. 1.5 28 ~d Parameters e1 and e3 allow additional flexibility in representing the plant population. Biologically, they could be related to adaptability to crowding (plasticity) or to initial planting arrangement. Regardless of the exact values of e1 and e3, some growth reduction and mortality increase will occur even (2.19) at low levels of crowding. This is reasonable for natural populations because the random distributi9n of seedlings places some plants unusually close to their neighbors and competition begins well before the total plot surface is used. Enhanced Model with Additional Constraints The basic model can be modified to incorporate other constraints on plant growth. Here, limitations on individual plant weight and total population biomass are added to the basic spatial constraint. It is assumed that approach toward any of the three constraints reduces the population's growth rate, but only the spatial constraint and the carrying capacity affect the mortality rate. Also, the deleterious effects of three constraints are assumed to be additive. The augmented model is dN ~ e e = -m N [(f Nl-2p 82p) 1 + (--B---) 2 J 8max (2.20) .. 29 and e e e ~~=gOB [l _ (f Nl-2p 82p) 3 _ (--B--) 4 _ ( B ) 5] 8max N wmax (2.21) where Bmax is the carrying capacity, wmax is the maximum individual weight, and e1 through e5 control how abruptly the rates respond to changes in the level of each constrained quantity. Model Analysis and Results Basic Model Although the basic model of equations 2.18 and 2.19 is derived from very simple assumptions, it can not be solved to give explicit equations for N(t) and B(t). However, it is amenable to isocline analysis, a technique discussed in many introductory ecology texts. This method is applied here to the simple case of e1 = e3 = 1 represented by equations 2.14 and 2.15. First, note that the growth rate of population biomass is zero when the right hand side (RHS) of equation 2.15 is zero, that is, g0 B [ 1 - f B2P Nl-2p ] = O ? On log transformation and algebraic manipulation, this yields 1 og B = 1 (- 2P + 1 ) 1 og N 1 2P log f , the equation of a straight line of slope 8 = -l/(2p) + 1 when (2.22) (2.23) 30 ordinate log B is plotted against abscissa log N. This slope must be negative or zero because 0 ~ p ~ 0.5, so that -l/(2p) ~ -1. Restrictions on the path of the model population in the log B-log N plane (the self-thinning curve) can be deduced from this zero isocline, which divides the log B-log N plane into two regions. Below the isocline, population biomass is increasing and trajectories move upward, while above the isocline, biomass is decreasing and trajectories move downward. Since dN/dt is strictly negative, the force of mortality is always decreasing density and moving the population leftward in the plane. Because the isocline slants up toward the left of the plane, it must intercept the downward and leftward path of any population starting above the isocline. Such a population steadily approaches the isocline, then crosses it to enter the lower half of log B-log N plane. A population below the zero isocline must move upward {dB/dt > 0), but can not grow through the isocline because dB/dt is zero along that line. Three possibilities remain: the population trajectory could approach the zero isocline asymptotically, remain a constant distance from the isocline, or move leftward faster than upward, thus moving away from the isocline. The potential ambiguity in the behavior of the model population below the zero isocline can be eliminated as follows. The instantaneous direction, st' of the population's path at any point in the log B-log N plane is given by d log B d log N = = St , (2.24} 31 a relationship that can be used to identify points of the log B-log N plane where the slope of the population trajectory is less than or equal to any given value ~' that is, where St ~ ~. Combining this inequality with equations 2.18 and 2.19 in the ratio of equation 2.24 yields go [ 1 - f Nl-2p s2P J -m f N1- 2P s2P < ~ . Algebraic manipulation and log transformation of this expression gives a relationship for log B in terms of log N: log B < (- 2~ + 1) log N- 2~ log f + 2~ log (g0g~ m~) ? (2.25) (2.26) The first two terms of the right hand side (RHS) of this equation simply give equation 2.23 for the zero isocline. The third term is a constant added to the intercept of the zero isocline since g0, m, and ~are constants. The equality in 2.26, which is the locus of points where trajectories take a given slope ~' defines a straight line parallel to the zero isocline. In fact, the zero isocline equation 2.23 is the special case of the more general equation 2.26 with ~ = 0. This general relationship can be used to find regions of the plane below the zero isocline where a population's instantaneous trajectory is steeper (more negative) than the slope of the zero isocline, so that the population is moving closer to the isocline. 32 Substituting ~ = S = -l/(2p) + 1 (the slope of the zero isocline) in equation 2.26 gives log B < (- 2~ + 1) log N - 1 log f 2p + 1 go ? 2P log(g 0- m[-1/(2p}+l]) (2.27) Since g0 > 0 and m > 0 while 0 ~ p ~ 0.5, the third term on the RHS is negative and equality defines a straight line parallel to but below the zero isocline. This lower line is asymptotically approached by all populations. Below this asymptotic trajectory, a population's path is steeper than the asymptotic trajectory, while the path of a population above the asymptotic trajectory is less steep than the asymptotic path. In both cases, the population's trajectory must continuously move closer to the asymptotic trajectory. Thus, the asymptotic trajectory has both attributes of? the self-thinning line: populations approach it from any starting point in the log B-log N plane, and it is a boundary between allowable and unallowable biomass-density combinations. This second conclusion follows because populations can never grow through the asymptotic trajectory from below, and even if the model was started above the thinning line, mortality and negative growth would drive the population trajectory toward the thinning line and out of the region of the plane representing untenable biomass-density states. Equation 2.27 also gives the slope and intercept of the self-thinning line in terms of the model parameters. The slope is -l/(2p) + 1, a function of the single parameter p which relates area 33 occupied to plant weight. Only if p = l/3 is the model thinning slope equal to the value of S = -1/2 predicted by the self-thinning rule. The thinning line intercept is given by the last two terms of equation 2.27 log K 1 1 go 7P log f + ~ log(g o- m[-17(2p)+l ]). (2.28) The value of log K depends on all the model parameters, but the first term of the RHS depends only on p and f. If the second term is small relative to the first, then log K is mainly determined by these two parameters. To determine when this condition is satisfied, define a new quantity, ~ log K, as the second term of ~quation 2.28 ~ log K = 1 g 0 2P log(g0 - m[-l/(2p)+l]) (2.29) which is the contribution to the intercept of the self-thinning line of the second term of equation 2.28 and is also the vertical distance between the asymptotic population trajectory and the isocline of zero biomass increase. Since g0 and m are both constants, m can be expressed as the product of g0 and a constant w, that is, m = w g0? With this substitution, equation 2.29 gives ~ log K in terms of p and w ~ log K = 1 1 2P log( 1 + w [2P - 1 ] ) (2.30) 34 This expression is negative or zero, so the thinning line is always below the zero isocline or equal to it. Figure 2.2 plots~ log K against log w for several choices of p and shows that most combinations of wand p give~ log K ~ 1. Only when pis small (p < 1/3) or w is large (w > 10) does ~ log K exceed one, and ~ log K is much less than one for most reasonable parameter values. Thus, log K values of four or more are primarily determined by f and p as specified by the first term of equation 2.28. Several features of this analysis are illustrated in Figure 2.3a, in which equations 2.14 and 2.15 are fitted to a Pinus strobus plantation remeasured nine times between 12 and 51 years after planting (lot 28, Spurr et al. 1957). By repeated trials, the parameter values g0 = 0.5, m = 0.0475, p = 0.29, and f = 0.006 were found to give a visually good fit to the log B-log N data when the model was started at the first data point and solved numerically using the LSODE differential equation solver (Hindmarsh 1980). The resulting solution demonstrates that the model can represent actual population data quite well. Isopleths of equation 2.26 for four values of ~?are also shown, including the zero isocline of biomass growth (log B = -0.724 log N + 3.83) given by~= 0, and the asymptotic self-thinning line (log B = -0.724 log N + 3.78) given by w = 8 = -l/(2p) + 1 = -0.724. The self-thinning line is -0.50 log units below the zero isocline, that is, ~ log K = -0.05. Figure 2.3b shows how model solutions for different initial states converge on the asymptotic self-thinning line. .. 0 -1 ~ ~ -2 0) 0 _J 0 were eliminated from the simulation (wi,t is the weight of plant at timet). The initial conditions for a simulation were specified by (3.9) selecting an initial number of plants and a location and weight for each plant. Initial weights for plants were chosen randomly from a normal distribution with mean wo and standard deviation swo' while plant locations were assigned from a uniform random distribution on a circular plot of specified area. The model was typically started with 200 plants, but 600 was the initial population size when the dynamics of the size distribution were of particular interest and larger sample sizes were needed to obtain accurate estimates of the moments of that distribution. Model Analysis The model was analyzed in three steps: (1) verification that the dynamics resemble real self-thinning behavior, (2) estimation of the variations in the self-thinning trajectory strictly due to stochastic model elements, and (3) evaluation of simulation experiments in which a parameter was varied among simulations to determine the effect of that parameter on self-thinning. A reference set of parameters was defined to generate the model 54 solutions used in steps one and two. The values chosen (Table 3.1) are arbitrary, but within biologically realistic limits. Except for special cases detailed in the results, the model solutions used here were linear over the interval 3.0 ~log N ~ 3.7, so regression of log B against log N was applied to data within this interval to estimate the slope, S, and intercept, a = log K, of a self-thinning line for each simulation. The intercept of each regression line at log N = 3.35, &3?35, was also calculated to provide a measure of th~nning line position near the center of range of data. Coefficients of determination were so high (all r2 > 0.97) that the regression estimates of S and & were identical to those from principal component analysis. PCA is preferable to regression when the two methods give different results (Chapter 4). The time required for the density of individuals to fall from the initial value of 10,000 to 1000 was recorded as a measure of the average rate of self-thinning. Because the model is complex, a controlled procedure was used to identify model parameters that affect the self-thinning trajectory. In each of six simulation experiments, a parameter was varied across 5-15 simulations and the resulting group of self-thinning trajectories was compared to a control group of ten trajectories generated by the reference parameter set. This control group was analyzed to estimate the ranges of variation in 8, &, and &3?35 strictly due to stochastic model factors. Values of S, a, or &3?35 well outside these ranges in an experimental .. 55 simulation indicated that the parameter being studied did alter the self-thinning line beyond the normal limits of stochastic variation and therefore has a role in positioning the self-thinning line. The seven parameters that have entries in the "Range of Variation" column of Table 3.2 were investigated in these simulation experiments, but one experiment considered parameters b and q together. For each experimental group, log B-log N plots were used to visually compare the variation among thinning trajectories to the variation among the control trajectories. The mean, range, and coefficient of variation (CV) of S, a, and a3.35 were also tabulated for each simulation experiment and compared to the same statistics for the control group. For five of the experiments, Spearman correlation coefficients, rs, (Sokal and Rohlf 1981) between the experimental parameter and 8, a, and a3.35 were calculated and tested for statistical significance to determine if these thinning line descriptors varied systematically with the experimental parameter. Although this application of statistical tests to simulation results may initially seem contrived, it is appropriate for a model with stochastic factors. Chance variations among a small number of stochastic simulations can produce meaningless correlations in the same manner as in real data. When high correlations were found, further analysis was done to estimate the precise relationship between the slope or intercept and the model parameter. 56 Table 3.2. Simulation Model Parameters and Their Reference Values. Reference Range of Symbol Meaning Units Value Variation p Allometric power relating 0.333 0.27- ZOI radius to weight 0.49 d Density of biomass in g;m3 6200 631- occupied space 39810 Competition algorithm 1-6 gl Maximum assimilation rate per g;m2/time 25 10-50 unit of ground area covered b Constant relation metabolic (g/m2)(1/q) 0.00147 1.47xlo-l0_ rate to weight 147* Power relating metabolic 2 * q 0.5-3.5 rate to weight Aplot Plot area m2 0.02 0.000632-0.0632** T Height to radius ratio for VOl's of plants of weight wo RGRmin Minimum survivable relative 0 growth rate ~t Time step time o. 1 (~t)m Time interval over which time 0.1 RGR's are averaged no Initial number of plants 200 wo Initial average weight g 0.0001 swo Standard deviation of g 5xlo-5 initial weight *b and q were varied together in a single set of simulations (see text). **Aplot was varied to give initial plant densities from 3160 to 316,000 plants;m2. 57 Results The simulation model mimics several behaviors of real plant populations (Figure 3.1). Initially uncrowded populations trace a nearly vertical path in the log B-log N plane, but eventually bend toward and move along a negatively sloped self-thinning line. Model populations show other behaviors of real plant monocultures, including nearly logistic increase in biomass {Hutchings and Budd 198la), approximately exponential mortality during self-thinning (Yoda et al. 1963, Harper 1977), and the development of skewed size distributions from initially symmetric distributions (White and Harper 1970, Hutchings and Budd 198la). Information on individual simulations is presented in Table 3.3, including the experimental' parameter value {if any) and the three thinning line descriptors S, a, and a3?35? All the thinning line regressions were based on at least eight points, but most used between 20 and 70. Coefficients of determination, r2, were uniformly high (r2 ~ 0.97), confirming that trajectories were well described by a straight line over the range 3.0 ~log N 5 3.7. The thinning line descriptors are further summarized in Table 3.4, where means, ranges, and coefficients of variation are given for each of six experimental or control groups, along with Spearman correlations with the experimental parameter (if any). Table 3.3 also gives the time required for the density to fall from 10,000 to 1000. Systematic changes in these times within the simulation experiments show that all parameters affected the (a) (b) (c) 10000 1200 ~ / 8000 E 1000 ~ N "'-. ~ E N "'-. rn E ~2 "'-. Vl 800 - Vl rn c 6000 \ Vl ~ 0 0 -E Vl a. (/) 600 ~ \ 0 0 \ ?- E >-m 0 - 4000 ?- Vl rn m 400 c 0 Q) _J 0 2000 200 0 0? 0, 2.5 3.0 3.5 4.0 0 20 40 60 80 100 120 0 20 40 60 80 Log,0 Density (plants/m') Time Time Figure 3.1. Typical dynamic behavior of the simulation model. (a) is a log B-log N self-thinning plot, (b) shows the logistic increase in total population biomass, and (c) shows approximately exponential mortality commencing with the onset of thinning around time 20. The reference parameters (Table 3.2) were used. 01 (X) 100 120 59 Table 3.3. Self-thinning Lines for the Simulation Experiments. Estimated Thinning Line Intercept at Parameter Parameter Time to Sl~e Intercept log"~ = 3.35 Varied Value log N = 3* A 1:$ a "3.35 ,. None 70 -0.69 4.94 2.64 (Figure 55 -0.55 4.48 2.63 3.2) 54 -0.55 4.46 2.63 67 -0.65 4.83 2.67 59 -0.62 4.73 2.66 58 -0.59 4.62 2.65 59 -0.72 5.05 2.62 59 -0.64 4.79 2.65 58 -0.54 4.45 2.64 60 -0.67 4.90 2.66 p 0.27 426 -0.98 6.69 3.41 (Figure 0.29 203 -0.91 6.23 3.17 3.3) 0.31 106 -0.69 5.22 2.91 0.33 65 -0.70 5.02 2.66 0.33333 57 -0.61 4.68 2.64 0.35 41 -0.57 4.42 2.50 0.37 27 -0.42 3.71 2.30 0.39 19 -0.35 3.36 2.18 0.41 14 -0.28 2.97 2.02 0.43 11 -0.24 2.70 1.89 0.45 8 -0.22 2.54 1.80 0.47 7 -0.09 2.00 1.68 0.49 6 -0.11 1.97 1.58 d 631 6 -0.45 3.17 1.65 (Figure 1000 9 -0.55 3.66 1.83 3.4) 1585 15 -0.56 3.90 2.03 2512 22 -0.60 4.25 2.26 3981 36 -0.60 4.47 2.45 6310 60 -0.61 4.70 2.65 10000 90 -0.63 4.93 2.83 15850 158 -0.63 5.16 3.06 25120 251 -0.55 5.11 3.27 39810 378 -0.59 5.44 3.45 Competition 1 59 -0.68 4.93 2.64 a 1 gorithrn 2 36 -0.70 4.76 2.43 (Figure 3 65 -0.64 4.60 2.47 3.5) 4 84 -0.57 4.65 2.7 3 5 >100 -0.62 4.89 2.82 6 44 -0.74 4.99 2.52 gl 10 >100 -0.54 4.43 2.63 (Figure 15 >100 -0.71 5.03 2.65 3.6a) 20 71 -0.64 4. 79 2.64 25 55 -0.59 4.60 2.62 30 52 -0.64 4.80 2.65 35 37 -0.63 4.76 2.63 40 38 -0.59 4.62 2.65 45 36 -0.71 5.06 2.68 50 30 -0.64 4. 79 2.65 Q 0.5 58 -0.62 4.71 2.64 (Figure 1.0 64 -0.64 4.80 2.64 3.6b) 1.5 59 -0.64 4.78 2.65 2.0 57 -0.58 4.59 2.64 2.5 61 -0.67 4.88 2.62 3.0 73 -0.68 4.92 2.65 3.5 57 *The time required for plant density to drop from its initial value of log N = 4.0 to log N = 3.0. Table 3.4. Thinning Line Statistics for the Simulation Experiments. Sl~pe Intercept Intercept at log N = 3.35 1\ 1\ a. 0.3.35 Parameter Figure Varied Mean Range cv r~ Mean Range cv r~ Mean Range cv r~ 3.2 none -0.62 0.18 10.2 4.72 0.59 4.5 2.64 0.05 0.6 3.3 p -0.48 0.89 61.2 0.99* 3.96 4.72 39.2 -1.oo* 2.37 1.83 24.4 -1.oo* (<0.0001) (<0.0001} ( <0.0001) 3.4 d -0.58 0.17 9.2 -0.49 4.48 2.28 16.3 0.99* 2.55 1.80 24.0 -1.oo* {0.15} {<0.0001) (<0.0001) 3.5 Competition -0.66 0. 16 8.9 4.80 0.38 3.3 2.60 0.39 5.9 en 0 algorithm 3.6a gl -0.63 0.17 8.9 -0.20 4.76 0.63 4.2 0.25 2.64 0.06 0.6 0.40 {0.61} (0.52) (0.28) 3.6b q -0.64 0.10 5.7 -0.60 4.78 0.34 2.5 0.60 2.64 0.03 0.4 0.26 (0.21) (0.21) (0.62) aspearman correlation of the thinning line descriptor with the model parameter. The significance level of the correlation is given in parentheses. *Significant at the 95% confidence level (P $ 0.05}. 61 rate of self-thinning, except q, which had no effect, and initial density, which was not amenable to this analysis. The log B-log N plot of the control group simulations (Figure 3.2) shows that there are variations in the self-thinning trajectory that can only be attributed to persistent effects of the stochastic initial conditions. However, the slopes of the self-thinning lines are quite similar and the lines are all positioned within a narrow vertical band. Only the allometric power, p, altered the self-thinning slope. This effect is demonstrated visually by comparing the experimental thinning diagram (Figure 3.3a) to Figure 3.2 and quantitatively by the high Spearman correlation between p and 8 (rs = 0.99, P < 0.0001). The relationship between 8 and 1/p is linear (Figure 3.3b), and the regression equation 8 = -0.555(1/p) + 1.05 (r2 = 0.98, P < 0.0001, 95% CI for slope= [-0.604, -0.506], 95% CI for intercept= [0.91, 1.18]) was very close to the ideal linear relationshipS= -0.50 (1/p) + 1.00 developed in Chapter 2. The parameter p also altered the self-thinning intercept, a, (rs = -1.00, P < 0.0001), which was also linearly related to (1/p) (Figure 3.3c), again as predicted in Chapter 2. The regression equation was a= 2.97(1/p} - 4.18 (r2 = 0.99, P < 0.0001, 95% CI for slope= [2.80, 3.13], 95% CI for intercept= [-4.65, -3.72]). The parameter d, the density of biomass in occupied space, affected the position of the self-thinning line (Figure 3.4a), as measured by the thinning intercept a (rs =0.99, p < 0.0001). 62 ............. N E "'-.. 0) 2 ....__, (/) (/) 0 E 0 OJ ~ 1 0) 0 _.J 3.0 3.5 4.0 Log 10 Dens it y ( p I an t s / m 2) Figure 3.2. Variations in the self-thinning trajectory due to stochastic factors in the simulation model. The ten runs all used the reference parameter set (Table 3.2) and differed only in the initial plant locations and weights. (a) Vl ~ 2 E 0 ?-CD , 0> 0 1 -' 0~.-----.----~~ 3.0 3.5 4.0 Log,0 Density (plants/m2) (b) < 65 The relationship between & and log d is apparently linear (Figure 3.4b), as predicted in Chapter 2. The regression equation is & = 1.186 log d + 0.0891 (r2 = 0.97, P < 0.0001). Thinning slope was not affected by d, as shown by the low correlation between d and 8 (rs = -0.49, P = 0.15) and by the fact the range and CV of S in the experimental group (0. 17 and 9.2%, respectively) were not greater than the corresponding values for the control group (0.18 and 10.2%). The thinning lines are more widely spread among the six simulations using the different competition algorithms (Figure 3.5) than among control group simulations, indicating that thinning line position is affected by this algorithm. Spearman correlations could not be calculated for this simulation experiment because the competition algorithm is a nominal variable that can not be ranked, but further support does come from a comparison of the ranges of variation of &3?35? This range is eight times larger among the experimental simulations than in the control group, and the CV is almost ten times larger. A similar comparison of & values seems to contradict this conclusion; however, when the interpretations &3?35 and & differ, a3?35 is the more reliable measure of thinning line position because it lies in the middle of the range of the data while & is an extrapolation of the regression lines well outside the data (Sokal and Rohlf 1981}. A lack of effect of the competition algorithm on thinning slope is indicated by the similar means and ranges for a in the control and experimental groups. ,...--.._ N E ~ 0) ....__,; en en 0 E 0 m S? 0) 0 _J 66 3 t\.._ ??? ???????????? 2 I' I I I I I I I I 0~--~------~--------~--------~------~~ 3.0 3.5 4.0 Log,0 Density (pI ants/ m 2) Figure 3.5. Effect of the competition algorithm on the self-thinning line. Curves for six simulations are are marked with competition algorithm identification numbers (Table 3.1). " 67 The assimilation rate, g1, had no effect on the self-thinning trajectory, as indicated by similarity of Figure 3.6a to Figure 3.2, the near equality of the ranges and CVs of the thinning line descriptors to the same statistics for the control group, and the low Spearman correlations of g1 with the thinning line descriptors. Metabolic cost parameters b and q were likewise unimportant in positioning the self-thinning line {Figure 3.6b). In this simulation experiment, the power q relating maintenance cost to weight was varied from 0.5 to 3.5. Simultaneously, the parameter b was adjusted so that the initial maintenance cost {second term of equation 3.1) for a plant of weight w0 {lo-4 g) was constant across the seven simulations. However, as growth increased average weight above w0, differences in maintenance cost due to differences in q became important. Interpretation of this simulation experiment is complicated by the two trajectories that are decreasing in both biomass and density by the end of the simulation. The explanation is that maintenance costs affect the maximum possible size for a model plant, which can be calculated by setting equation 3.6 equal to zero and solving for wmax to obtain = (3.10) The differences in q across the simulations led to differences in wmax and this limitation became important in two simulations; however, all the simulations followed a common path through the log B-log N plane until the two populations became limited by (a) (b) 3 (c) 3 3 .--.. N .--.. .--.. E N "'-E E Ol ~~ "'- "'-Ol Ol 2 VI 2 '--" 2 .,__.. VI VI VI 0 \~ VI VI E \ ~ 0 0 0 E E ?- 0 0 (D ?- ?- " 1 (D (D Ol 0 S! 1 0 Ol Ol _J 0 0 _J _J 0 3.0 3.2 3.4 3.6 3.8 4.0 3.0 3.5 4.0 3 4 5 Log,0 Density (plants/m 2) Log 10 Density (pI ant s/m 2) Log 10 Dens. it y (pI ant s/m 2 ) Figure 3.6. Simulations for three parameters that did not affect the self-thinning line. In (a) parameter gl was varied from 10 to 50 g/m2/time unit across ten simulations. (b) was generated by using six values of parameter q between 0.5 and 3.0. Two curves are marked with the associated values of q (Table 3.4). (c) shows thinning trajectories for five simulations started at initial densities of 3160, 10000, 31600, 100000, and 316000 plants;m2. .. 0'1 co 69 wmax and diverged sharply from the common path. Before fitting the thinning line in the q = 3.0 simulation, data points with log N values below 3.4 were eliminated. No thinning line was fitted to the q = 3.5 simulation since the maximum weight limitation took effect before the linear portion of the thinning trajectory was established. The visually evident lack of effect of q on the thinning line estimates is further supported by the failure of a, &, and &3?35 to exceed the limits established in the control group and by the low Spearman correlations between q and the thinning line descriptors. Variations in initial density also had no effect on the position of the self-thinning line (Figure 3.6c}. The five different initial densities were created by placing 200 plants on different sized plots. This was the only way to vary initial density over a large range because model limitations precluded direct manipulations of the initial number of plants. The maximum initial number was 600 and simulations were stopped when less than 20 plants remained because the thinning trajectory made undesirably sharp jumps at low population sizes. Although the resulting trajectories start from different points in the plane, all converge on the same thinning line, at least within the limits of the stochastic variation seen in Figure 3.2. A more quantitative analysis was not possible because there was no common range of linear behavior over which to fit self-thinning lines, so this simulation experiment is not included in Tables 3.3 and 3.4. 70 Discussion This analysis has identified only one parameter, the power p relating area occupied to individual weight, that affected the slope of the model self-thinning line. Three parameters affected thinning line position: p; the density of biomass in occupied space, d; and the competition algorithm. These results agree with the simpler model of Chapter 2 and together suggest that the regularities of self-thinning can be explained by the shape-dependent occupation of space. The linearity and slope of the self-thinning line are determined by the power relationship relating space occupied to plant size, and the thinning intercept is related to at least two additional factors. The implications of these results have already been discussed in Chapter 2. The possibility that the conclusions of Chapter 2 were biased due to an over-simplified, spatially homogenous model can now be discarded because the detailed representation of individual size, location, and competitive interactions in the simulation model led to the same results. Although spatially averaged models are not useful for many applications in plant ecology (Schaffer and Leigh 1976), they are appropriate tools for investigating self-thinning because of the unusual degree of spatial uniformity that is present in even-aged monospecific stands. Model parameters representing resource availability and utilization efficiency (maximum assimilation rate g1) and 71 metabolic costs {q and b) affected only the rate of self-thinning, not the slope or position of the thinning line. Initial density also did not affect the thinning line. However, the model did not allow the parameters of individual shape and density, p and d, to vary in response to these important environmental factors. Real plants can respond to environmental limitations by varying their shapes {Hutchings 1975, Harper 1977) and canopy densities {Lonsdale and Watkinson 1983a), and density stress can cause permanent alterations in plant geometry {Peet and Christensen 1980). The exact responses would vary among species. Environmental factors could, then, induce changes in the self-thinning line, not because the factors are key determinants of the thinning line, rather because they affect the thinning line indirectly by altering the growth parameters of the plants. The important effects of the competition algorithm on the rate of self-thinning and the position of the thinning line are significant new results of this analysis. The competition algorithm is related to the allowable overlap between plants {Chapter 2) and to the degree of asymmetry in competitive interactions, so measures of these factors should be related to the self-thinning line. This prediction is tested in Chapters 5 and 6, where the relationships of shade tolerance with thinning slope and intercept are considered. The persistent effect of the stochastic initial conditions seen in Figure 3.2 is also interesting. Since all parameters were held constant, these differences could only be attributed to random variations in the initial weight distributions and initial plant 72 locations. This result suggests that even if all other sources of variation could be eliminated from thinning experiments, observed self-thinning lines could still differ in position because of these stochastic factors. Some results of this analysis are relevant to other theories about the causes of the self-thinning rule. The unimportance of metabolic parameters b and q in positioning the self-thinning line argues against the hypothesis that the self-thinning rule derives from a 2/3 power relationship between metabolic costs and plant weight (Jones 1982), and against an important role in fixing the self-thinning slope for the fraction of assimilate devoted to maintenance (Pickard 1983}. The simulations of Figure 3.5 indirectly address another theory that the -3/2 exponent of the w-N relationship is maintained during thinning by mutual adjustment of plant allometry and stand structure (the distribution of individual sizes--Mohler et al. 1978}. Figure 3.7 shows how the frequency distributions of individual plant weight change with time for the four simulations generated by competition algorithms 1, 2, 3, and 5. These time plots of the first four moments of the weight distribution (average, coefficient of variation, skewness, and kurtosis) show that the four competition algorithms lead to widely different dynamics of stand structure. If the mutual adjustment hypothesis is correct, thinning slopes should also be different since stand structure varies without any possibility of compensatory adjustments in plant allometry (parameter p was 1/3 for all four 73 (a) 4,...---------------, .... .r::. Ol ?-CI> 3: 2 (c) Cl> Ol 0 .... Cl> > <( .... .r::. Ol ?-Cl> 3: ..... 0 rn rn Cl> c 3: Cl> .::,(. Vl 0 3 2 0 -1 -2 -3 0 50 100 150 200 Time 2 0 50 100 150 200 Time (b) .... .r::. 0.8,---------------, Ol ?-Cl> 3: ..... 0 0.6 c 0 .... 0 .... 0 0.4 > ..... 0 .... c Cl> 0.2 ?-u ..... ..... Cl> 0 u 0.0 -1...,----,------,---,.----'""T""" 0 50 100 150 200 Time WI -2~--~---,---,.----~ 0 50 100 150 200 Time Figure 3.7. Dynamics of the weight distributions in four simulations with different competition algorithms. Algorithm numbers (Table 3.1) were 1 (solid line), 2 (dashed line), 3 (dotted line), and 4 (chain-dotted line). The mean, coefficient of variation, skewness, and kurtosis of each of the weight distributions are shown in (a) through (d). Methods of calculation are given in Sakal and R oh 1 f ( 1981) ? 74 simulations). However, if plant allometry sets the thinning slope, the simulations should give the same thinning slope despite the differences in competitive regime and size distribution dynamics. Since thinning slope did not vary among the four simulations, the allometric hypothesis is supported and the mutual adjustment hypothesis is not. " 75 CHAPTER 4 SOME PROBLEMS IN TESTING THE SELF-THINNING RULE Introduction Many self-thinning lines with slopes near y = -3/2 have now been reported, and their sheer number is considered strong evidence for the self-thinning rule, or even a self-thinning law (see references in Hutchings 1983). However, an analysis of size-density data is not simply a matter of regressing log w against log N and so demonstrating the self-thinning rule. Some important analytical difficulties must be fully discussed before the large body of evidence is embraced as convincing proof for the rule. The principal problem areas are (1) the data selected to test the hypothesis, (2) the points used to estimate a thinning line, {3) the . curve fltting methods used, (4) the choice between the log ~-log N or log B-log N formulations of the rule, and (5) the conclusions drawn from the results. Recommendations for resolving some difficulties are presented here, and the implications for acceptance of the self-thinning rule are discussed. Selecting Test Data Many data sets have been reported to exhibit a linear relationship between log w and log N with a slope near y = -3/2, but it is usually easy to find evidence for a hypothesis, regardless of whether or not it is generally true. Therefore, the mere 76 existence of such evidence does not verify the hypothesis. Rigorous verification instead comes from failure of the opposite endeavor, to find evidence that contradicts or falsifies the hypothesis (Popper 1963). The emphasis on compiling corroborative evidence for the thinning rule has diverted attention from data that do not conform. Violations of the rule have been discussed only for shoot populations of clonal perennials (Hutchings 1979) and for thinning under very low illumination (Westoby and Howell 1982, Lonsdale and Watkinson 1982), but there are other violations besides these special cases, such as thinning slopes of y = -2.59 and -4.5 for tropical trees Shorea robusta and Tectona grandis (O?Neill and DeAngelis 1981), y = -1.2 the temperate tree Abies balsamea (Sprugel 1984), and y = -3.2 for seedling populations of the woodland herb Allium ursinum (Ernst 1979). Information contradicting the self-thinning rule has been missed even in the very sources from which supporting evidence has been drawn. For example, data for one stand of Pinus strobus (Spurr et al. 1957) have been repeatedly cited in self-thinning studies (Hozumi 1977, 1980, Hara 1984), and a self-thinning slope of y = -1.7 has been fit (White 1980). However, the report of Spurr et al. also presented data for a second stand which gives a thinning slope of y = -2.11 (Table A.2). This second, unreported stand contradicts the self-thinning rule in two ways: the thinning slope is quite different from the predicted value, and both the slope and intercept change from stand to stand. A second example refers to a yield table for Pinus ponderosa (Meyer 1938) reported to give a .. ? 77 thinning line of log w = -1.33 log N + 4.06 {White 1980). However, the log B-log N thinning plot for the complete yield table {Figure 4.la) shows that 13 thinning lines could be fit since information is given for 13 values of site index, a general measure of site including soil composition, fertility, slope, aspect, and climate (Bruce and Schumacher 1950). The existence of the twelve unreported thinning lines contradicts two tenets of the self-thinning rule: the thinning line is not independent of site quality and the thinning intercepts are not species constants~ The data from some yield tables even give different thinning slopes for different site indexes, as shown in Figure 4. lb for Sequoia sempervirens {Lindquist and Palley 1963). A second major problem in testing the self-thinning rule arises because the thinning line is an asymptotic constraint approached only as stands become sufficiently crowded. To estimate the slope and position of a thinning line using linear statistics, data points from populations that are not limited by the hypothesized linear constraint must be eliminated. These would include points from young populations that have not yet reached the thinning line, older stands understocked because of poor establishment or density-independent mortality, and senescent stands. Failure to eliminate such points will bias the thinning line estimates (Mohler et al. 1978), but when the data are confounded by biological variability and measurement errors, recognition and elimination of (a) 5~--------------------------------~ ............ N E "'-.. 0> '---"' Vl Vl 0 E 4 0 Cil E Q) +- (/) Q 0> 0 _J 3 -2 -1 0 Log 10 Density (plants/m 2) (b) Vl Vl 0 E 0 Cil E Q) +- (/) Q 0> 0 _J 5.0 4.5 4.0 3.5 3.0 2.54---,---~-.--~---r--~--~--~--~ -1.4 -1.3 -1.2 -1.1 -1.0 Log 10 Density (pI ants/ m 2) Figure 4.1. Two forestry yield tables showing variation in thinning line parameters with site index. Dotted lines connect data points and solid lines are PCA thinning lines. (a) shows data for Pinus ponderosa (Meyer 1938). Thinning slopes, 8, for ten site indexes were near -0.31 (between -0.307 and -0.326), but intercepts ranged from 3.75 to 4.18. (b) shows data for Sequoia sempervirens (Lindquist and Palley 1963). Thinning slopes for six site indexes ranged from -4.15 to -1.81 while intercepts ranged from -1.93 to 2.67. Tables B.l and B.2 give additional information. 79 spurious points is difficult. Since there is no ~priori estimate of the thinning line position, decisions to eliminate data points must be made~ posteriori (Westoby and Howell 1982}. This is true even if detailed field notes (Mohler et al. 1978) or mortality curves (Hutchings and Budd 198la) are available to aid the process. With such ~ posteriori manipulations, no thinning analysis can be done in a strictly objective way. Figure 4.2 presents three data sets that illustrate these problems. Each plot shows how the slope, intercept, r2, significance level, and confidence interval all change with the points used to fit the thinning line (Table 4.1). The results are very sensitive to certain points, yet there is no objective way to decide whether or not to include those points. The sensitivity of thinning line parameters to the choice of data points also has important statistical implications. The uncertainties about including or excluding some points should be counted in forming confidence intervals and performing tests of significance. However, existing statistical methods do not take such uncertainties into account, so estimated r 2 values are too high and confidence intervals are too narrow. Some data sets show more that one region of linear behavior and so present the analyst with still another subjective decision: Which linear region is relevant to the self-thinning rule? Figure 4.3 presents a yield table for Populus deltoides (Williamson 1913) that illustrates this problem. White (1980) fit the thinning line log w = -1.8 log N + 3.08 through the data for ages 7 to 15, while (a) ~ N E ...__,_ 0> ---- (/) Ill 0 E 0 ?- m -0 0 .s:; (/) t? 0 --' 4.0 --- .. 3.8 3.6 3.4? 3.2 '~: 0 ' D 0 -1.5 -1.0 -o.5 o.o ?o.5 Log,. Density (plants/m2) (b) 3.0,---------------. ,..... 'E ...__,_ 2.8 ~ Ill ., E 2.6 0 m -; 2.4 0 .s:; (/) .. g' 2.2 --' .c 0 10 0 0 2.0+-----.------,---...,..---..J 2.0 2.5 3.0 3.5 Log,. Density (pI ant s/m2) (c) Ill Ill 0 g 2.5 ?-m -0 0 .s:; (/) 5! 2.0 0> 0 --' 8 3.0 3.5 Log,. Density (plants/m2) Figure 4.2. Three examples of the sensitivity of self-thinning line parameters to the points chosen for analysis. In each plot, several thinning lines (capital letters) are fitted to different combinations of points (numbers) as indicated in Table 4.1. Data in (a), (b), and (c) are for Populus tremuloides (Pollard 1971, 1972), Triticum sp. (Puckridge and Donald 1967, White and Harper 1970), and Tagetes patula (Ford 1975). 0 0 4.0 (X) 0 81 Table 4.1. Three Examples of the Sensitivity of the Fitted Self? thinning Line to the Points Chosen for Analysis. Points Figurea Lineb Includedc r2 pd 4.5a A l-2 1.00 B 1-3 0.94 0.16 c l-4 0.94* 0.030 0 2-4 0.99* 0.010 4.5b A l-10 0.63* 0.0061 B 1-3,5-10 0.86* 0.0003 c 5-10 0.91* 0.0030 D 6-10 0.98* 0.0008 E 1-3,5-9 0.80* 0.0027 F 1-3,5-8 0.66* 0.027 4.5c A 1-10 0.34 0.075 B l-8 0.06 0.54 c 3-10 0.48 0.056 D 4,6,9, 10 0.84 0.079 PCA Thinning Linee Slope Intercept 95% CI a -0.19 3.73 -0.25 3.66 -0.26 [-0.48,-0.07] 3.65 -0.53 [-0.64,-0.42] 3.72 -0.20 [-0.33,-0.08] 3.32 -0.24 [-0.32,-0.15] 3.43 -0.31 [-0.46,-0.18] 3.70 -0.36 [-0.45,-0.28] 3.84 -0.16 [-0.25,-0.08] 3.26 -0.13 [-0.24,-0.02] 3.17 -0.41 4.36 +0.24 2.22 -0.70 5.38 -0.35 4.03 aspecies names and references are given in the legend of Figure 4.2. bThese letters label the fitted lines in Figure 4.2. CThe numbers of data points in Figure 4.2 used to fit the thinning line. dThe statistical significance of the log B-log N correlation. eThe method of fitting the thinning line by principal component analysis is discussed in Chapter 5. *Significant at the 95% confidence level (P ~ 0.05). 82 ......-... N 4.2 E "-.._ 0> '--"' (/) 4.0 (/) 0 E -- 0 m 3.8 E Q) +- (/) 52 3.6 0> 0 ~ 3.4~--~----------~--------~------------~ -2.0 -1.5 -1.0 -0.5 Log 10 Density (pI ants/ m 2) Figure 4.3. Example of a data set with two regions of linear behavior. White (1980) fit the dotted line log w = -1.80 log N + 3.08 to yield table data (solid line) for juvenile stands (7-15 years old) of Populus deltoides (Williamson 1913). Older stands follow the dashed line log B = -0.20 log N + 3.85 fit to 25-50 year old stands (Table 8.2) 83 the present study has fit a line of shallower slope, log B = -0.20 log N + 3.85, to the data for ages 25 through 50. The line fit by White may be more typical of the steep ascent of juvenile populations through the log B-log N plane, while the second line may represent the self-thinning behavior of more mature stands. Fitting the Self-thinning Line Most self-thinning lines have been estimated by linear regression of log w against log N, but regression is inappropriate for thinning data because log N is not a good independent variable, that is, it is neither measured without error nor controlled by the experimenter. Although principal component analysis (PCA--Mohler et al. 1978) and geometric mean regression (GMR--Gorham 1979) have been proposed as more appropriate fitting methods, the true self-thinning line is actually not estimable. A In general, the slope, Byx' of a linear bivariate relationship for Y in terms of X is given oy the linear structural relationship (Madansky 1959, Moran 1971, Jolicoeur 1975), ~ syy - A sxx + ~yx = -- where sxx' s , and s are sums of squares and cross products YY xy corrected for the mean and A is the ratio of the error variance in Y to the error variance in X. These error variances are the 84 residuals around the true linear relationship between Y and X, not the total sample variances. Particular values of A give the standard methods as special cases of this general equation: A = ~ (no error in X) gives Byx = sxyisxx' the regression of Y against X; A = 0 (no error in Y) gives Byx = syylsxy' as obtained from regressing X against Y; A= 1 (equal marginal variances of X and Y) gives the PCA solution; and A = syy/sxx (marginal variances ~n the same ratio as the total sample variances) gives the GMR solution, Syx = sign(sxy) syylsxx? If the relationship between X and Y is strongly linear (high r 2), then all solutions are similar, but as the association becomes less strict (lower r2), the discrepancies among the solutions increase and results become more sensitive to A. Figure 4.4 shows these four particular solutions for a typical thinning data set (Mohler et al. 1978). Although this data set showed a very significant log B-log N relationship (r2 = 0.40, P < 0.0001), the four solutions ranged from 8 = -0.34 to -0.87, and the regression of log B against log N gave -0.34 while PCA gave -0.41. Two sources of variation are reflected by A: measurement errors, which can be estimated by replication, and natural biological variability, which can not be estimated (Ricker 1973, 1975). Since A can not be known, the true solution of equation 4.1 is unestimable (Ricker 1975, Sprent and Dolby 1980) and the slope of the self-thinning line must be based on some assumed value of A. For most self-thinning data, there is no basis to (/) (/) 0 E 0 CD 0 +- 0 I- S2 0'> 0 _J 85 4.5,----------------------------------------- 4.0 3.5 3.0 A A ~ 2.5 2.0~~------~------~--------T-------~--~ -0.5 0.0 0.5 1.0 1.5 Log 10 Density (pI ants/ m 2) Figure 4.4. Thinning lines fit to one data set by four methods. The data are for Prunus pensylvanica (Mohler et al. 1978}. Lines were fit to the 34 square points by regression of log B on log N, regression of log N on log B, geometric mean regression, and principal components analysis {lines A, D, C, and B, respectively). Equations for the four lines were, respectively, log B = -0.34 log N + 3.88, log B = -0.87 log N + 4.19, log B = -0.55 log N + 4.00, and log B = -0.41 log N + 3.92, while confidence intervals for the slopes were [-0.50, -0.19), [-1.58, -0~60], [-0.72, -0.42], and [-0.61, -0.23]. For all four lines, r = 0.40 and P < 0.0001. 86 assume the marginal variation of log N is less than that of log B (or vice versa), so the most reasonable assumption is that the two marginal variances are approximately equal, giving A= 1 and leading to the PCA solution of equation 4.1. Regression analysis implicitly assumes that one of the variables is error free and A takes one of the extreme possible values A = 0 and A = w (Moran 1971). Since this extreme assumption is clearly untrue for self-thinning data, many reported thinning lines estimated by regression analysis are potentially in error. The use of PCA has created yet another statistical problem: many studies now report the percentage of variance explained (%EV) by the first principal component (PC) rather than the correlation coefficient, r, or r2? Unlike r2, which ranges from 0 to 1, %EV ranges from 0.5 to 1 because the first PC always explains at least 50% of the total variation, even if the two variables are completely uncorrelated. Therefore, %EV values are always higher than r2 values. This has been misinterpreted by some authors as an indication that PCA is more reliable then regression. Actually, the exact method of fitting a straight line (Y-X regression, X-Y regression, PCA, or GMR) is irrelevant to the calculation and interpretation of correlations or coefficients of determination since these measure the strength of linear association rather than the position of any particular line in the plane. The reported measure of association should always be r or r 2, regardless of the fitting method used (Sprent and Dolby 1980). 87 Choosing the Best Mathematical Representation The recommended use of PCA focuses attention on another question: Should the thinning rule be tested by relating log w to log N or by relating log B to log N? The two choices are mathematically equivalent (Chapter 1), and when the regression of log B against log N is compared to the regression of log w on log N for a set of data, the slopes differ by exactly one (8 = y + 1) and the confidence intervals have identical widths. However, neither of these conditions holds when log w-log N PCA is compared to log B-log N PCA, so one CI may include the predicted slope of the self-thinning rule while the other CI does not. Although mathematically equivalent, the log w-log N and log B-log N formulations of self-thinning rule are not statistically equivalent when appropriate curve fitting methods are used, so an explicit decision is required: Which formulation is more appropriate for testing the self-thinning rule? The log B-log N formulation is the correct choice because the log w-log N alternative suffers from two major limitations. Changes in average weight can be misleading because average weight increases when small individuals die, even if the survivors do not actually gain weight (Westoby and Brown 1980). Average size increases through two processes--growth of living plants and elimination of small plants--so that the average size of the stand increases more rapidly than the sizes of individuals composing it (Bruce and 88 Schumacher 1950). Attempts to relate log w to log N are actually correlating some combination of growth and mortality with mortality, so the results are difficult to interpret. However, total stand biomass only increases through growth, so a correlation of log B with log N directly addresses the growth-mortality relationship and focuses attention on the extent to which mortality permits a more than compensatory increase in the size of the survivors. The second shortcoming of the log ~-log N analysis is more serious and damaging to the case for the self-thinning rule: the analysis is statistically invalid, gives biased results, and leads to unjustified conclusions. To understand why, consider the methods used to measure plant biomass. Stand biomass is often measured directly by harvesting all the plants in a stand and weighing them as a single group. Even when each individual is weighed (or individual weights estimated from a relationship between weight and some plant dimension) stand biomass is still estimated directly as the sum of the individual weights. Average weight is then derived from the original stand measurements by dividing the biomass by the density. In general, there are "serious drawbacks" in analyzing such derived ratios (Sokal and Rohlf 1981), but these problems are particularly acute when the ratio is correlated with one of the variables from which it was derived. Such an analysis gives correlations that hav~ been variously called "spurious" (Pearson 1897, as cited in Snedecor and Cochran 1956), ??artificial" (Riggs 1963), and "forced" (Gold 1977). ?? 89 A high correlation between log w and log N is both unsurprising and meaningless because log N is used to calculate log w. This is most easily seen when the original measurements of log B and log N are unrelated so that the sample correlation coefficient is low and not statistically significant. After deriving average weight from log w = log B - log N, log w is a function of log N even though log B was not. The variance of log w is Var(log B) + Var(log N) - 2 Cov(log B,log N) (Snedecor and Cochran 1956), which reduces to Var(log B) + Var(log N) because the covariance of unrelated variables is zero. Thus, the variance of log w is higher than the variance of log B, and all of the additional variation is directly attributable to log N. Since log N explains more of the variance in log w than in log B, the correlation between log w and log N is higher and more significant than the correlation between log B and log N. Although mathematically real, this higher correlation does not represent an increase in the information content of the data, but is a 11Wonderful tool for misleading the unwary .. (Gold 1977). The deceptive effects of this data transformation are also present in simple plots of the data. This is important because self-thinning data must be edited to remove extraneous points before fitting a thinning line. Since plots of the data are essential tools for recognizing such points, a data transformation that creates artificial linear trends in the plot will obviously disrupt the editing procedure. In the most extreme case, the distorted log w-log N plot may suggest a linear relationship when none was present in the original log B-log N data. More subtle errors arise 90 when a real log B-log N correlation exists, but points that are not associated with the constraint of the thinning line are mistakenly included in fitting the thinning line. A few examples will illustrate how the deceptive effects of the data transformation pervade the existing evidence for the self-thinning rule. In the first example, the log w-log N plot (Figure 4.5a} for a study of Trifolium pratense (Black 1960) shows a linear trend and high correlation (r2 = 0.76) between log w and log N among nine data points purported to form a thinning line that agrees with the self-thinning rule (White and Harper 1970}. However, the untransformed log B-log N plot (Figure 4.5b) shows the true situation: there is no significant negative correlation between log B and log N for the nine points, and time trajectories of stands cut steeply across the proposed thinning line rather than approaching it asymptotically. Figure 4.6 illustrates a less extreme case where a linear trend is present, but the log w-log N plot gives a distorted impression of which points lie along the constraining line. The log B-log N plot of the data (Chenopodium album, Yoda et al. 1963) shows an apparent linear constraint which could be estimated by fitting a line through the 13 square data points in Figure 4.6b. The remaining 14 points are distant from the constraint and should be removed before curve fitting, but this is hidden in the distorted log w-log N plot (Figure 4.6a), where the artificial linearization causes the extraneous data points to fall in line with the others. Fitting a thinning line through all 27 data points by PCA of log w against .. (a) (b) 3.0 ,......._ Ol ~ -0.6 ,....... N -+- E ~ ...c "-. 4 rn rn I Q) ~ 2.5 ' 3 VI VI -+- -0.8 2. 0 lj 0 2 E 0 0 ...c (/) m Q) -+- 2.0 rn -1.0 0 0 0 L ..c Q) (/) > <( 5! rn 5! 0 1.5 Ol -1.2 _J 0 _J 3.25 3.50 3.75 4.00 2 3 4 Log 10 Density (plants/m2) Log 10 Density (plants/m 2) Figure 4.5. Example of a spurious reported self-thinning line. (a) White and Harper (1970} fit the solid regression line log w = -1.33 log N + 3.86 (n = 9, r2 = 0.76, P = 0.003) to data for Tr~folium pratense (Black 1960}. In (b) the nine points are in the box and other points omitted by White and Harper are also shown. The original measurements of log B and log N for the nine points are uncorrelated (r2 = 0.13, P = .33). Time trajectories for different initial densities (dotted lines) cut through the proposed thinning line and do not approach it. 1.0 _. (a) (b) 4 0 ,-.... 0> 0 ,-.... ....._., "' ' E -...c 0 ?? .. oo " 0> ?o 0> 0 '?, ....._., Q) ' 0 3: ' (/) 0 C:l? ??? o (/) 0 ~ 0 0 0?. E - o??. 0 0 0 o ??~o m I- 0 Q) ' ??,o 0 0> 0 -1- 0, -0 0 L I-Q) ',,_o > ' 12 <( ()'., 0> 12 0 0> _J 0 -2- ' ',,C? _J 3 3 4 5 3.0 3.5 4.0 4.5 5.0 Log 10 Density (pI ants I m 2 ) Log 10 Density (pI ants I m 2) Figure 4.6. Example of potential bias in data editing in the log w-log N plane. Data are for Chenopodium album (Yoda et al. 1963). In the log w-log N plot (a) all 27 data points follow a common linear trend, represented by the dotted line. The equation of this line, estimated from ~CA of log Band log N, is log B = -1.33 log N + 3.94 (r2 = 0.90, P < 0.0001, 95% CI for S = [-1.53, -1.16]). When the artificial enhancement of linearity is removed on examination of the log B-log N plot of (b), an apparent linear constraint is still evident; however, many data points fall relatively far from the constraining line (triangles). A new (solid) PCA line through the 13 points closer to the border of the constrained region has the equation log B = -0.41 log N + 5.15 (r2 = 0.93, P i 0.0001, 95% CI for 6 = [-0.48, -0.33]), and the inadequacy of the dotted line is revealed. 1.0 N 93 2 log N gives log w = -1.33 log N + 3.94 (r = 0.90, P < 0.0001, 95% CI for slope= [-1.53, -1.16]}, while the 13 points actually near the constraint in the log B-log N plot give log B = -0.41 log N + 5.15 (r2 = 0.93, P < 0.0001, 95% CI for slope= [-0.48, -0.33]}. The distorted log w-log N plot changes the selection of relevant points, the estimated thinning Jine, and its comparison to the self-thinning rule. The transformation can artificially straighten data that are actually curved in the log B-log N plane. The straight lines in the log w-log N plane of Figure 4.7a seem to be a reasonable fits to the data (Fagopyrum esculentum--Furnas 1981}, but the log B-log N plot (Figure 4.7b} reveals the true curvature of the data and the inadequacy of the straight line model. This curve straightening deception is important in editing data sets with juvenile or senescent stands, which often curve gradually toward or away from the thinning line. A final example shows how the log w-log N plot can lead to questionable conclusions about the effects of an experimental treatment on self-thinning. To evaluate the effects of a fertilizer treatment on self-thinning behavior, five plots of Erigeron canadensis received different fertilizer applications in a ratio of 5:4:3:2:1 before seeds were planted (Yoda et al. 1963}. The conclusion that the thinning trajectory was insensitive to soil fertility seems justified in the log w-log N plot (Figure 4.8a}, where the five treatments seem to approach the same thinning line despite the large differences in fertility. However, the (a) -..c 0> Q) ?; 0 0 - -1 0 1- Q) 0> 0 I... Q) > <( ~ 0> -2 0 _.J 0 0 0 0 0 0 0 3.0 3.5 4.0 4.5 Log 10 Density (pI ants/ m 2 ) (b) (I) (I) 0 E 0 m 0 -0 3.0 I- 2.5 ~ 0> 0 _.J 0 3.0 0 0 0 0 0 0 0 0 0 3.5 4.0 4.5 Log 10 Density (pI ant s/m 2) Figure 4.7. Example of deceptive straightening of curves in the log w-log N plane. Data are from an experiment with Fagopyrum esculentum where the circle population received five times as much fertilizer as the square population (Furnas 1981). Both trajectories appear reasonably linear in the log w-log N plot (a), and the two lines log w = -1.50 log N + 4.837 and log w = -1.50 log N + 4.622 reported by Furnas seem to fit the data well. The log B-log N plot (b) reveals the the inadequacy of the straight line model for these data. (a) 0,-------------------------------~ -+- ..c t:J) -1 Q) 3: 0 +- 0 -2 1- Q) t:J) 0 !.... Q) > -3 <( S! t:J) 0 .....J -4 ~. ?--,_~."' ?---, '"' '?,,, '"' ',, '"' 3 '"' '"'?"' '"'.. ...... ? .. '1 . '?, 4 5 Log,0 Density (pI ants I m 2) (b) ,......._ 2.5 N E '-... t:J) -._.; (I) (I) 0 E 0 m 0 +- 0 1- S! t:J) 0 .....J 2.0 1.5 1.0 3 4 5 Log 10 Density (pI ants I m 2) Figure 4.8. Potential misinterpretation of experimental results due to deceptive effects of the log w-log N plot. Numbers marking data points (Erigeron canadensis--Yoda et al. 1963) indicate the relative amount of fertilizer applied before planting. Yoda et al. concluded that the overall trend of the data show a slope of -1.5 (chain-dot line) in the lo~ w-log N plot (a). White (1980) reported the dotted regression line log .w = -1.66 log N + 4.31 (r = 0.93, P < 0.0001, 95% CI for slope= [-1.86, -1.47]). In (b) the present study fit the PCA line (solid line) log B = -1.04 log N + 5.70 (r2 = 0.99~ P <-0.0001, 95% CI for slope= [1.12, -0.96]). 1.0 (.]1 96 log B-log N (Figure 4.8b) plot suggests a very different interpretation. The populations do respond to fertility differences and initially follow very different trajectories, but the differences gradually disappear and are gone around the fourth harvest, about 4.5 months after planting. It seems reasonable that over this long interval, leaching and plant uptake removed fertilizer from the soil, all five treatments approached a background fertility level, and only then did the treatments converge on a common trajectory. The log B-log N plot also suggests a thinning line that is very different from the expected value of a= -l/2. This plot reveals a fact that is obscured in the log w-log N plot: all five plots lost biomass between harvests two and three, which is not surprising because these harvests were made during the winter. Because winter conditions apparently interfered with growth, early harvests should not be included in estimating a thinning line. PCA analysis of harvests four through six gives log B = -1.04 log N + 5.70 (r2 = 0.99, P < 0.0001, 95% CI for slope= [-1.12, -0.96]). This slope is not close to the hypothesized value of a= -1/2 and is statistically different from -1/2 (P < 0.0001). Testing Agreement with the Self-thinning Rule Evaluation of the self-thinning rule has been hindered by the lack of an objective definition of how close to the predicted value a thinning slope must be to agree quantitatively with the rule. White (1980) reported that many thinning slopes fall between ? 97 y = -1.8 andy= -1.3 and presented all these values as examples of the same quantitative rule. However, this arbitrary range has no objective basis and its limits represent very different predictions about population growth. A population following a thinning trajectory of slope y = -1.3 over a 100-fold decrease in density will increase its biomass about fourfold, while a population following a trajectory of slope y = -1.8 will increase its biomass about 40-fold over the same 100-fold density decrease. Although the two populations show qualitatively similar behavior in following a linear trajectory in the log size-log density plane, they show a tenfold difference in biomass response to the same degree of density decrease and thus would not seem to obey the same quantitative rule. Over a 1000-fold density decrease, one population shows an eightfold biomass increase, while the other thinning population shows a 250-fold biomass increase and the discrepancy between the two thinning regimes increases exponentially as larger amounts of - Q) Q) (.) 0 20 c >- 15 (.) c c 15 Q) (.) I.. Q) Q_ Q) c :::l IT Q) 10 Q) u !.... !.... 15 u... Q) Q_ Q) :::l IT Q) !.... 10 u... 10 10 5 5 5 5 0 m n rTTl rT n 0 lhl lliln _o 0 -5 -4 -3 -2 -1 0 0 2 4 6 8 10 Thinning Slope~ Thinning Intercept a (c) 70 (d) 45 40 140 60 200 35 120 50 30 Q) Q) 0 40 c Q) (:) I.. 30 Q) Q_ >- Q) 100 >-(.) Q) u c 0 25 c Q) Q) :::l c 80 :::l IT Q) IT Q) (.) 20 Q) !.... !.... !.... u... Q) 60 u... Q_ 150 100 15 20 40 50 10 10 5 20 0 n n--rJ 0 -10 -8 -6 -4 -2 0 -6 -4 -2 0 2 4 6 8 Thinning Intercept A a Thinning Slope~ Figure 5.1. Histograms for the slopes and intercepts of fitted thinning lines. (a) and (b) show the distributions of slope and intercept, respectively, for log B-log N thinning lines in the experimental and field data. (c) and (d) show the same distributions for thinning lines in the forestry yield table data. <'" 113 Table 5.2. Comparisons of Thinning Line Slope and Intercept Among Plant Groups. Group n 1\ Slope S Mean Median Experimental and Field Data Herbaceous monocots 8 -0.44 -0.39 Herbaceous dicots 25 -0.74 -0.65 Temperate angiosperm trees 15 -0.65 -0.53 Temperate gymnosperm trees 19 -0.87 -0.65 Eucalyptus trees 4 -1 .26 -1 .03 Tropical angiosperm trees 4 -2.56 -2.55 All of the above 75 -0.85 -0.62 Intercept & Mean Median 4.45 4.24 5.17 5.09 3.78 3. 72 3.79 3.88 2.87 3.07 2.20 2.21 4.18 3.97 Tests for Significant Differences Among Six Groups A NOVA Kruskal-Wallis F(5,69) = 7.68 F(5,69) = 10.9 H(5) = 17.9 H(5) = 41.1 Forestry Yield Table Data Temperate angiosperm trees 58 Temperate gymnosperm trees 281 Eucalyptus trees 12 A11 temperate trees 339 All of the above 351 -0.60 -0.80 -3.90 -0.77 -0.88 -0.63 -0.61 -4.39 -0.61 -0.62 p < 0.0001* p < 0.0001* p = 0.0031* p < 0.0001* 3.50 3.54 1.09 3.53 3.45 3.56 3. 72 l. 79 3.68 3.68 Tests for Differences Between Gymnosperms and Angiosperms A NOVA Kruskal-Wallis F(l,337) = 4.85 F ( 1 , 337) = 0. 13 H(l) = 3.77 H(1) = 8.30 Tests for? Differences Among Three Groups A NOVA Kruskal-Wallis F(2,348) = 79.9 F(2,348) = 38.9 H(2) = 14.9 H(2) = 11.9 p ::: 0.028* p ::: 0.8 p = 0.052 p = 0.004* p < 0.0001* p < 0.0001* p = 0.0006* p = 0.0027* *Significant at the 95% confidence level (P ~ 0.05) 114 two groups while the Kruskal-Wallis analysis did. This indicates that the mean values of & are similar, but one of the groups has a disproportionate of number extreme values of & (in this case, the gymnosperm group has more high values), causing the rank based Kruskal-Wallis test to detect a difference. Thinning slope and intercept were both correlated with shade tolerance in the FYD. The correlation analysis was done separately for the angiosperms and gymnosperms because of the observed differences in S and a between these two groups. The Eucalypt group was not analyzed because shade tolerances were not available. For the 46 angiosperm thinning trajectories analyzed, ? was significantly correlated with shade tolerance but thinning intercept was not lTable 5.3). Both 0 and & were significantly correlated with tolerance in the gymnosperms; however, the sign of the correlation of S with shade tolerance was opposite to that observed for the angiosperms, further justifying the separate analyses of the two groups. Thinning line parameters were not constant for species considered in more than one study. Both S and & show considerable variation within species in the EFD (Table 5.4). In some cases, parameter estimates for a given species are quite different, but the confidence intervals for the estimates are so large that the differences are not statistically significant. In other cases, the differences are statistically significant, as indicated by non-overlap of the 95% confidence intervals. Eight A (20%) of the 40 possible pairwise within-species comparisons of S 115 Table 5.3. Spearman Correlations of Shade Tolerance with Thinning Line Slope and Intercept in the Forestry Yield Data. Means for Shade Tolerance Groupsa Thinning Parameter 2 3 4 Temperate Angiosperms 1\ Slope S -0.391 -0.547 -0.685 (10) (18) (18) Intercept & 3.632 3.437 3.517 (10) (18) (18) Temperate Gymnosperms 5 1\ SlopeS -0.916 -0.748 -0.642 -1.149 -0.459 (32) (78) (47) (69) (41) Intercept & 3.123 3.438 3.732 3.280 4.172 (32) (78} (47) (69) (41) asample sizes are given in parentheses. bstatistical significance level. Spearman Correlation -0.52* (46) -0.19 (46) pb 0.0002 o. 22 0.35* <0.0001 ( 267) 0. 57* <0. 000 -~ (267) *significant at the 95% confidence level (P ~ 0.05). Table 5.4. Thinning Line Slopes and Intercepts of Species for which Several Thinning Lines Were Fit from Experimental or Field Data. Thinning Line Parametersa Slope Intercept I d. Codeb Conditione r2 s 95% CI Diff.d A 95% CI Diff. d (l Abies sachalinensis A l9A 0.841 -0.649 [-0.776,-0.535]* c 4.16 [ 4.15, 4.17] c B 24T 0.839 -0.465 [-0.965,-0.104] c 4.39 [ 4.07, 4.83] c C ll9A 0.971 -2.786 [-5.645,-1.772]* AB l. 71 [-0.49, 2.49] AB ...... Beta vulgaris ...... 0'1 A 33T 0.592 -1.335 [-3.355,-0.648]* 6.38 [ 4. 54' 11 ? 80 J B 34T 0.645 -2.304 [-5.478,-1.348]* CDEF 9.93 [ 7. 08' 19. 38] CDEF c 43A 18% L. I. 0.957 -0.662 [-0.839,-0.509]* B 4.79 [ 4. 17' 5.50] B D 43A 25% L. I. 0.916 -0.692 [-0.973,-0.470] B 5.12 [ 4.22, 6.25] B E 43A 37% L.I. 0.940 -0.668 [-0.886,-0.486] B 5.09 [ 4.39, 5.94] B F 43A 55% L. I. 0.934 -0.649 [-0.838,-0.487] B 5.22 [ 4.59, 5.95] B G 43A 100% L. I. 0.698 -0.648 [-1.415,-0.197] 5.30 [ 3.55, 8.27] ? ? " Table 5.4. (continued) Thinning Line Parametersa Slope Intercept Id. Codeb Conditione r2 8 95% CI Diff.d 1\ 95% CI Diff.d (). Erigeron canadensis A 15T 0.930 -0.621 [-0.688,-0.558]* B 4.36 [ 4.19, 4.55] B B 21T 0.987 -1.038 [-1.121,-0.962]* A 5.70 [ 5.43, 6.00] A Eucalyptus regnans ...... A 98A S.I. 28.9 0.971 -2.478 [-5.012,-1.559]* 1.39 [-1.63, 2.48] ...... --.J B 98A S.I. 33.5 0.964 -1.066 [-2.132,-0.549]* 3.44 [ 2.31, 3.99] Lo1ium perenne A 38A 100% L. I. 0.549 -0.324 [-0.674,-0.034] 3.79 [ 2.88, 4.89] B 91A 100% L. I. 0.908 -0.427 [-0.543,-0.319] 4.80 [ 4.37, 5.27] c 91T 100% L. I. 0.854 -0.245 [-0.330,-0.163]* 4.20 [ 3.87, 4.54] 0 92A 23% L.I. o. 776 -0.544 [-1.509,-0.011] 4.33 [ 2.28, 8.06] E 92A 44% L. I. 0.786 -0.503 [-1.273,-0.027] 4.28 [ 2.48, 7.20] Table 5.4. (continued) Thinning Line Parametersa Slope Intercept Id. Codeb Conditione r2 " 95% CI Diff.d " 95% CI Diff. d B a. Picea abies A 137A 0.983 -0.422 [-0.462,-0.383]* 3.90 [ 3.88, 3.92] B 137T 0.982 -0.433 [-0.476,-0.392]* 3.97 [ 3.95, 3.99] Pinus strobus _. ...... A 8A Lot 2B 0.986 -0.724 [-0.830,-0.628]* B 3.78 [ 3.74, 3.83] BC 00 B 8A Lot 2C 0.987 -1.116 [-1.278,-0.976]* A 3.34 [ 3.22, 3.43] A c 93A 0.955 -0.954 [-1.189,-0.764]* 3.44 [ 3. 25, 3.60] A Pinus taeda -- A 82A 0.468 -0.305 [-0.499,-0.130]* B 4.21 [ 4.11, 4.30] B B 102A 0.939 -0.670 [-0.837,-0.526]* A 3.42 [ 3.23, 3.59] A ~ " Taole 5.4. (continued) Thinning Line Parametersa Slope Intercept Id. Codeb Conditione r2 s 95% CI Diff. d II a 95% CI Diff. d A lOA Full light 0.836 B 35A 0.716 -0.473 -0.622 Trifolium subterraneum [-0.660,-0.310] [-0.928,-0.382] 4.60 [ 3.97, 5.17 [ 4.33, 5.33] 6.23] aonly species for which more than one statistically significant thinning line (P ~ 0.05) were fit are included (see Table A.2 for actual significance levels). bTable A.l associates the numeric part of each Id. code with a particular species and study. The letter indicates the type of biomass measurements made: A = aboveground parts only, T = aboveground and belowground parts both included. csee Table A.l and the references given there for more information on condition. dThe letters in this column identify table entries (see letters at left of table) for the same species that are significantly different from the current estimate of S or &. Statistical disagreement (at P ~ 0.05) between estimates is indicated by non-overlap of confidence intervals. *Indicates slopes that are significantly different (P ~ 0.05) from the value B = -l/2 predicted by the self-thinning rule. __, __, \,0 120 were significant, as were 9 (23%) of the 40 possible comparisons of &. Since the confidence level is 95%, 2 of 40 comparisons should be different by chance alone, but both analysis gave a least 4 times this number. For the FYD, individual thinning line estimates (Table B.2) were not retabulated because so many thinning trajectories (264) were involved and because 95% confidence intervals could not be calculated. Instead, the sample sizes, means, standard deviations, coefficients of variation, minima, maxima, and ranges are given for S and & for each species (Table 5.5). Although significance tests were not possible, the large observed ranges again indicate that a and ~ are not species constants. Discussion The predictions of the self-thinning rule have been tested here through five different analyses: (l) statistical tests of the hypothesis that the slope of the self-thinning line isS = -1/2, (2) examination of the frequency distributions and ranges of variation of thinning line slope and intercept, &, (3) tests for variation in @ and & among plant groups, (4) tests for correlations of S and ~with shade tolerance, and (5) examination of the variation in ~and a for particular species. Although many data sets do show the predicted region of linear association between log B and log N, all five analyses indicate that the slope of this relationship does not take the same value S = -l/2 across the entire plant kingdom. Therefore, the evidence .. .. ? Table 5.5. Statistics for Thinning Line Slooe and Intercept of Species for which Several Thinning Lines Were Fit from Forestry Yield Data. Number ofb 1\ 1\ Thinning Slope s Thinning Intercept a Yield Thin. Species a Tables Lines Mean soc cvd Min. Max. Range Mean soc cvd Min. Max. Range Species Considered in More Than One Yield Table - ---- -- --- --- Alnus rubra 3 8 -0.41 0.19 46 -0.65 -0.12 0.53 3.46 0.42 12 2.69 4.08 l. 39 __, N Picea glauca 2 9 -0.75 0.30 39 -1.28 -0.52 0.76 3.59 0.31 9 3.01 3.83 0.82 __, Pinus banksiana 3 10 -0.79 0.39 50 -1.73 -0.24 1.49 3.06 0.27 9 2.51 3.51 1.00 Pinus ech1nata 4 20 -0.72 0.23 32 -0.97 -0.40 0.56 3.53 0.35 10 3.14 4.05 0. 91 Pinus elliotti 2 11 -0.56 0.16 28 -O.H3 -0.38 0.45 3.62 0.34 9 3.22 3.99 0.77 Pinus palustris 2 14 -1.04 0.22 21 -1 .28 -0.80 0.48 3.01 0.49 16 2.20 3.52 1.32 Pinus ponderosa 3 27 -0.72 0.66 92 -2.44 -0.31 2.13 3.36 0.90 27 1.08 4.18 3. 10 Pinus resinosa 2 8 -1 .07 0.48 45 -1.97 -0.63 1.34 3.46 0.47 14 2.73 4.21 1.48 P1nus stroous 3 12 -0.74 o. 18 25 -1 .07 -0.53 0.54 3.68 o. 15 4 3.43 3.88 0.45 Pinus taeda 3 17 -0.69 0.20 28 -0.90 -0.25 0.65 3.48 0.34 10 3.05 4.10 1.05 Pseudotsuga menziesii 3 20 -0.60 0.15 25 -0.71 -0.26 0.45 3.80 0.22 6 3.59 4.26 0.67 Sequoia semoervirens 2 10 -2.68 1.45 54 -4.75 -1.22 3.53 1.20 2.24 186 -1.93 3.44 5.37 Tsuga heteroPhylla 4 37 -0.44 0.08 18 -0.61 -0.36 0.25 4.21 0.14 3 3.03 4.42 0.48 Table 5.5. (continued) Number ofb 1\ /\ Thinning Slooe 8 Thinninq Intercept a Yield Thin, Species 8 Tables Lines Mean soc cvd Min. Max. Range Mean soc cvd Min. Max. Range Species Considered~ More Than One Yield Table Abies balsamea 1 4 -0.59 0.00 1 -0.60 -0.59 0.01 3.80 0.02 1 3. 77 3.81 0.05 Abies concolor 1 7 -0.57 0.01 2 -0.59 -0.56 0.03 3.84 0.13 4 3.67 3.98 0.31 Castanea dentata l 3 -0.65 0.01 2 -0.66 -0.64 0.02 3.62 0.06 2 3.56 3.67 0. l1 Chamaecyoaris thyoides l 6 -0.53 0.01 2 -0.54 -0.51 0.03 3.80 0. 14 4 3.53 3.89 0.36 Eucalyptus globus l 4 -5.80 1.86 32 -8.13 -3.80 4.33 1.06 0.98 93 -0.21 2.06 2.27 Eucalyotus microtheca l 3 -6.84 0.39 6 -7.22 -6.44 0.78 -3.67 0.51 14 -4.18 -3.17 1.01 N Eucalyptus sieberi 1 3 -0.62 0.07 12 -0.69 -0.55 0.15 4.12 0.03 1 4.09 4.14 0.06 N Liriodendron tulipifera l 3 -1.15 0.12 10 -1.24 -l .02 0.23 2.69 0. 18 7 2.53 2.88 0.35 Liquidambar styraciflua l 6 -0.39 0.03 8 -0.42 -0.33 0.09 3.87 0. ll 3 3.74 4.03 0.30 Picea mariana l 3 -0.83 0.04 5 -0.85 -0.78 0.07 3.72 0.01 0 3.72 3.72 0.01 Picea rubrens l 5 -0.54 0.00 l -0.55 -0.54 0.01 3.86 0. l 0 3 3.72 3.97 0.25 Pinus monticola 1 4 -0.79 0.02 2 -0.80 -0.77 0.04 3.88 0. ll 3 3.75 4.01 0.25 Populus tremuloides l 4 -0.33 0.07 23 -0.43 -0.27 0.16 3.62 0.12 3 3.46 3.74 0.28 Thuja occidentalis 1 6 -0.35 0.04 ll -0.41 -0.30 0. ll 3.58 0.08 2 3.46 3.67 0.21 3 0nly results from monsopecific yield tables are included (Tables B. 1 and B.2). hGives the number of yield tables for the species and the total number of thinning lines fit. cstandard deviation. dcoefficient of variation expressed as a percentage. 123 does not support the hypothesis that all plants obey the same quantitative self-thinning rule. These analyses also confirm predictions about the variations in thinning parameters from previous studies and from the models presented here. The statistical tests of the 63 thinning lines reported to demonstrate the self-thinning rule show that this body of evidence does not strongly support the rule. Many of the reported high correlations between log w and log N proved spurious when the log 8-log N data were reanalyz~, and 30% of the thinning lines did not even show a significant relationship between biomass and density. There could be two basic explanations for this result: the data are too variable or too few to detect the existence of a true relationship between log Band log N, or there really is no relationship, as when stand biomass is maintained at a carrying capacity. Either way, the data do not support the self-thinning rule. Seventy percent of the thinning lines did show the predicted significant linear relationship between log B and log N, but 32% of the thinning slopes were significantly different from S = -1 and disagreed quantitatively with the thinning rule. Deviations of the thinning slope from the predicted value are particularly important s because 8 is the exponent of a power relationship, B = K N so small differences in B represent large differences in the predictions of the equation (Chapter 4). Only 38% of the thinning slopes were both significantly different from 0 and not significantly different from B = -1/2. Because of the inherent ambiguities of statistical testing, these represent two 124 possibilities: a true relationship of slope 8 = -1/2 is present, or some different true slope is present but the data are so variable as to obscure this difference. In fact, many data sets were too variable to be useful in resolving alternative hypothesis about the slope of the thinning line. Strictly speaking, the most that can be claimed is that these 38% of the thinning lines do represent statistically significant relationships between log Band log Nand the slopes are close to 8 = -l/2 within the resolution of the data. In short, 30% of the data sets did not demonstrate any relationship between biomass and density while another 32% disagreed quantitatively with the -1/2 slope predicted by the thinning rule. This gives 62% of the data sets that were either useless for testing the rule or in quantitative disagreement with it, and only 38% that potentially support for the rule. These tests must be interpreted with an important caveat: the true confidence level of each test is less than the nominal 95% because the necessary step of editing the data to fit the thinning line increases the probability of a Type I statistical error (rejection of a true null hypothesis) by some unknown amount. Some additional percentage of the thinning lines actually showed no correlation between log B and log N, while some slopes that tested as significantly different from 8 = -1/2 are actually different at some lower confidence level. At present, all we can do is acknowledge and decry this limitation and argue that the analysis is still the most objective that is currently possible. Since the .. 125 other four analyses also show that thinning slopes vary widely from 8 = -l/2, confidence in the test results seems justified. The frequency distributions of thinning slope and intercept show greater variations in these parameters than suggested by the II " often cited ranges -0.8 ~ S ~ -0.3 and 3.5 ~ a ~ 4.4 proposed by White {1980). Both distributions have single modes near their accepted values of 8 = -1/2 and a= 4, but more extreme II values are also present. The tendencies for S to be near -l/2 and for & to be near 4 reflect the fact that values near the mode of a distribution are more frequently observed than extreme values. The commonness of modal values does not indicate that a limited range is rigidly imposed by a biological law. Seventy percent of the 426 " B values in the combined EFD and FYD were within White's range, as were 42% ~f the & values, so these ranges do include a large percentage of the observations, but not all. The greater variation observed here has a simple explanation, White's ranges were based on 36 data sets collected precisely because their thinning slopes. were close to B = -1/2, but attempts were made here to avoid prejudicing the analysis with this criterion. The observed departures of the thinning slope from S = -1/2 for trees in both the EFO and FYD support Sprugel's (1984) speculation that a thinning slope of y = -3/2 (S = -1/2) is the exception rather than the rule for woody plants. Since herbaceous species also showed such departures, Sprugel's prediction applies for all plants. White (1981) has suggested that thinning slopes steeper than y = -2 (S = -1) are prima facie evidence of 126 significant departure from the classic thinning rule. Seventeen (23%) of the EFD and 45 (13%) of the FYD slopes meet this criterion. Additional evidence of variation in thinning line parameters was found in the existence of statistically significant differences in thinning line parameters among plant groups, and in the observation of significant correlations between thinning line parameters and shade tolerance (as predicted by Westoby and Howell 1981, Lonsdale and Watkinson 1983a). These results argue against a single, quantitative thinning rule for all plants (as claimed by White 1981, Hutchings 1983). Thinning line slope and intercept were not even constant among thinning lines for a particular species, so these parameters are not species constants (as suggested by Mohler et al. 1978, Hozumi 1980, White 1981, Hutchings 1983). This result complements evidence that thinning slope and intercept can vary within a single yield table when data from different site indexes are compared (Chapter 4). This has also been reported by Hara (1984), and Furnas (1981) has shown experimentally that the position of a thinning line can respond to changes in nutrient availability. Unfortunately, the biological interpretation of a is confounded because thinning slopes are variable. The constants of power equations have direct biological interpretations only if the powers of the relationships are identical (White and Gould 1965). The existence of variation in thinning slope and intercept supports the the models of Chapters 2 and 3, which predicted that the slope of the thinning line depends on how the shape of the space occupied by a plant changes with growth, while the intercept is ? 127 additionally affected by the density of biomass in occupied space and the way interacting plants partition areas of overlap. If these models are correct, then thinning slope should vary among species because different species have different shapes and densities of biomass per unit of space (Mohler et al. 1978, Furnas 1981, Lonsdale and Watkinson 1983a). Thinning parameters should also vary within a species because plants change shape and canopy density in response to growing conditions (Harper 1977). Between-species and within-species variations are both present in the data analyzed here, just as predicted. The observed correlations of thinning line parameters with shade tolerance also confirm the importance of differences in allo~able overlap between plants in positioning the thinning line. 128 CHAPTER 6 SELF-THINNING AND PLANT ALLOMETRY Introduction The documentation of significant variation among the slopes of self-thinning lines (Chapter 5) verifies one major prediction of the mathematical model {Chapter 2), but the true test of the model comes in determining if the variations in self-thinning slope can be related to differences in plant allometry. Here three questions are addressed that provide such tests: (1) Are thinning slope and intercept correlated with available measures of plant allometry? (2) Do differences in thinning slope among plant groups {Chapter 5) reflect corresponding differences in plant allometry? (3) Do the correlations of thinning slope with shade tolerance (Chapter 5) reflect corresponding correlations between shade tolerance and plant allometries? The results and conclusions are related to those of previous studies to explain why allometric models have been discounted (Westoby 1976, Mohler et al. 1978, White 1981). Expected Relationships between Thinning Slope and Plant Allometry The model predicts that the slope of the self-thinning line in the log 8-log N plane is B = -l/[2p] + 1 (or equivalently y = -l/[2p] in the log w-log N plane), where p is the allometric power relating the radius of the zone of influence (ZOI) to weight according to the power equation R ~ w0 (Chapter 2). This 129 prediction is referred to here as the allometric hypothesis. Hypothetically, it could be tested by measuring both S and ~ for several populations and testing statistically to determine if e and p follow the expected relationship. However, the actual measurement of ZOis is problematic. It is difficult to define precisely where the ZOI of an individual ends, especially in highly competitive situations where the zones may be tightly packed and overlapping to varying degrees. Even when the ZOI is defined as the canopy area (White 1981), the actual measurement is still difficult and less accurate than other common measurements, such as height or bole diameter at breast height (DBH). Consequently, few data are available for relating the ZOI to weight and no published data were found where both ~ and ~ could be estimated simultaneously. Plant ecologists and foresters routinely measure some parameters of plant shape, such as height, DBH, or bole basal area (BSLA), that can be used to fit allometric equations. Thinning slope can be related to these other measures of plant allometry. For example, the allometric equation n ~ w~hw could be fit to average height and average weight measurements of the same stands A used to estimate a self-thinning slope. Values of ~hw for .A several species could then be correlated with S to test if the two parameters are significantly related. Geometric models can be further analyzed to suggest a functional form for the S-~hw relationship. Assume that plant height, the density of biomass in occupied space, and the ZOI radius all vary with plant weight according to allometric power functions 130 h ~ w~hw, d ~ w~dw, and Roc wP. If the volume of space occupied by a plant (VOl) is approximately cylindrical, then the volume is v = TI R2 h and plant weight is w = v d = TI R2 h d. Since w oc R2 h d and R, h, and d are allometrically related to weight, the two sets of equations can be combined to give w oc w2P w~hw w~dw and the allometric powers are constrained by 2P + ~hw + ~dw = (6? 1) The predicted relationshipS = -1/(2p) + 1 can be solved for p to give 1 = "'2 -r( ....... , ----::S~) (6.2) where the symbol pest emphasizes that p is not directly measured A but derived by mathematical transformation of measured S values. Combining equations 6.1 and 6.2 gives a relationship between Pest' ~hw' and ~dw that should obtain if the allometric hypothesis is true {6.3) This relationship is useful because of its linear form, simple geometric derivation, and clear representation of the compromises inherent in plant growth: allocation of more resources to height growth (higher ~hw) or to packing more biomass in space already occupied (higher ~dw) leaves fewer resources for expanding the 131 ZOI radially (lower p). Less radial expansion in turn means less conflict with neighoors, less self-thinning, and a steeper (more negatively sloped) thinning line. To focus on the compromises between height growth and radial growth, assume that d is fairly constant (~dw = 0). Equation 6.3 simplifies to 1 1 Pest=- 2 ~hw + 2 ' which defines a triangular region in the Pest-~hw plane that is bounded by the two axes Pest = 0 and ~hw = 0 and the by the line Pest= -0.5 ~hw + 0.5. Empirical measurements of Pest A and ~hw would lie around this line, except that ~dw is not (6.4) generally zero because the density of biomass in occupied space does vary with plant size (Lonsdale and Watkinson 1983a). Since measurements of d through time are not generally available for A estimating ~dw' equation 6.3 can not be used and variation in ~dw will contribute to the errors in the simpler model of equation 6.4. Ignoring ~dw and fitting a linear relationship A between p and ~hw will produce a line in the Pest-~hw , plane below equation 6.4; however, a negative correlation between A Pest and ~hw will still support the allometric hypothesis, particularly if the functional form of the relationship approximates equation 6.4. Other measurements besides height are commonly available for trees and can be used to fit allometric relationships to weight (DBH ~ w ~Ow or BSLA ~ w ~Bw) that can be related to s. However, 132 DBH and BSLA are measures of the tree bole, not the VOl, so the expected relationship of ~Ow or ~Bw can not be simply derived from the geometry of the VOl. Also, mechanical considerations require that bole diameter increase to support any increase in tree weight, whether the growth is upward, radial, or simply an increase in the density of biomass in the VOl. Hence the DBH-weight or BSLA-weight allometries should not be be sensitive measures of change in the VOl shape and should not necessarily correlate strongly with Pest? DBH and height data can be combined to fit another allometric relation h ~ DBH ~hD. Although interpretation of ~hD is again confounded because DBH is not a measure of the VOl, h is a VOl dimension so some expectations for the Pest-~hD relationship can be deduced geometrically. If the VOl expands only radially (p = 0.5 and ~hw = 0 in equation 6.4), then ~hD will be zero since height is constant. If the VOl expands both radially and upward (P < 0.5 and ~hw > 0), then DBH will increase to support the additional weight and ~hD will be positive. Thus, a line representing the relationship in a p-~hD plot would pass through the point (~h0,p) = (0,0.5) and would be negatively sloped. An observed negative correlation between Pest and $hD would, then, support the allometric hypothesis. Furthermore, BSLA is simply related to DBH by BSLA ~ DBH 2, so the above logic also applies for the relationship between pest and ~hB" Although the functional form of the relationship of Pest to ~hD or ~hB is not geometrically obvious, it is reasonable to use a linear model unless this assumption proves inappropriate. 133 Methods Information to test the allometric hypothesis was available in the sources of experimental and field data (EFD) and forestry yield table data (FYD) analyzed in Chapter 5 since average stand height, DBH, or BSLA were often reported along with stand biomass and density. All height and DBH measurements were converted to common units of m, while basal areas were converted to m2 and divided by the density of individuals to give average basal area per tree. Many studies did not measure one or more of these dimensions, so fewer data were available for estimating allometric relationships. Of the 75 data sets in the EFD that showed significant relationships between log B and log N (Chapter 5), 31 had accompanying ?measurements of stand height, 8 reported DBH, and 29 reported BSLA. Of the 351 EFD thinning lines, 325 had height measurements, 334 had DBH, and 318 had BSLA. Dimensional measurements from the same stands used to estimate ~ were log transformed and analyzed with principal component analysis (PCA, Chapter 5) to fit allometric equations relating log h, log DBH, and log BSLA to log w; log h to log DBH; and log n to log BSLA. For each proposed allometric relationship in the EFD, the null hypothesis that the two variables were uncorrelated was tested and only relationships significant at the 95% confidence level were retained for further analysis. Similar statistical tests for the FYD were not possible (see Chapter 5) so the EFD and FYD were again analyzed separately. 1~ A Values of 8 {Chapter 5) were transformed to pest values and correlated with the five allometric powers using both Spearman and Pearson correlation calculations. The correlation coefficients were tested and when a significant relationship was found, linear regression was used to estimate an equation for the relationship. Principal component analysis was not used because the objective was to predict pest from an independent allometric parameter rather than to estimate a functional relationship between two variables. With this objective, regression is the appropriate technique despite errors in the independent variable {Ricker 1973, Sakal and Rohlf 1982). Univariate descriptive statistics and histograms were calculated for each allometric parameter, and ANOVA and Kruskal-Wallis tests of differences among plant groups were performed. Spearman correlations of shade tolerance with each allometric power for the temperate angiosperm and temperate gymnosperm groups of the FYD were also calculated. Additional information on the groups and methods is given in Chapter 5. Results Table A.3 presents allometric powers fitted to data sets in the EFD. The PCA slope and intercept of each allometric equation are given, along with the correlation and statistical significance of the relationship. For the five allometric powers, &hw' iDw' ~ A 6 ~Bw' ~hD' and ~hB' there were 23, 8, 23, 6, and 19, respectively, statistically significant relationships. Table B.3 135 gives the results of allometric regressions for the forestry yield data. Values of Pest calculated from fitted thinning slopes (Chapter 5) are included in Tables A.2 and 8.2. Tables 6.1 and 6.2 give univariate statistics for pest and the five allometric powers in the EFD and FYD, while Figures 6.1 and 6.2 present the corresponding histograms. Two of the correlations between the transformed thinning slope and the allometric powers of the EFD were statistically significant (P < 0.05--Table 6.3, Figure 6.3). Values of ~hw were negatively correlated with Pest (r = -0.55, P < 0.0026) and 30% of the observed variation in Pest was explained by the regression Pest= -0.710 $hw + 0.501. This equation does not differ significantly in either slope or intercept from the predicted equation p = -0.5 ~hw + 0.5. The failure of $0w and ~hD to correlate significantly with Pest is inconclusive because of the small sample sizes {n ~B). The allometric power $8w also showed no significant correlation with Pest' despite a larger sample size, but BSLA was not not expected to be a good indicator of thinning slope. The allometry $hB showed a statistically significant correlation with p t (r = -0.44, P = 0.032) and the es regression explained 19% of the variation in Pest? All five correlations between pest and the allometric powers in the FYD were significant {P < 0.0001, Table 6.3), but the high confidence levels were partly due to the large sample sizes (n > 309). The coefficients of determination give a more realistic assessment of the ability of these allometric parameters 136 Table 6.1. Statistical Distributions of Allometric Powers for Experimental and Field Data. Allometric Parameter Statistic I~ '$ow $Bw "' ~hB Pest hw <~>ho n 75 28 7 28 6 24 Mean 0.298 0.317 0.341 0.795 1.070 0.401 Std. dev. 0.075 0.068 0.038 0.108 o. 134 0.118 cv (%) 25 21 11 14 13 29 Std. err. 0.009 0.013 0.015 0.020 0.055 0.024 Skewness -0.75 -0.31 -0.53 -0.15 -0.57 0.07 Kurtosis 0.444 -0.52 0.66 0.23 -0.44 -1.13 Range 0.332 0.262 0.118 0.503 0.355 0.386 Percentiles 0 (Min.) 0.104 0.184 0.274 0.542 0.860 0.229 1 * * * * * * 5 0.130 0.189 * 0.589 * 0.229 10 0.194 0.202 * 0.651 * 0.241 25 0.252 0.269 0.321 0.704 0.959 0.273 50 (Median) 0.308 0.321 0.347 0.827 1 .078 0.399 75 0.341 0.377 0.374 0.866 1.202 0.490 90 0.385 0.398 * 0.893 * 0.573 95 0.410 0.425 * 1.003 * 0.608 99 * * * * * * 100 (Max.) 0.436 0.446 0.392 1.045 1.215 0.615 *Indicates percentiles that are identical to the minimum (or maximum) values due to small sample size. 137 Table 6.2. Statistical Distributions of Allometric Powers for Forestry Yield Table Data. Allometric Parameter Statistic fl. .A $Bw ~hD A Pest q>hW? q>Dw q>hB n 351 325 334 318 323 309 Mean 0.298 0.274 0.368 0.747 0. 770 0.387 Std. dev. 0.067 0.059 0.057 0.108 0.230 0.130 c.v. (%} 22 22 16 14 30 34 Std. err. 0.004 0.003 0.003 0.006 0.013 0.007 Skewness -1.46 0.352 0.626 1.19 1.12 1.77 Kurtosis 2.91 0.666 11 .4 18.8 1.99 5.74 Range 0.392 0.399 0.644 1.341 1.435 0.927 Percentiles 0 (min.) 0.054 0.118 o. 165 0.330 0.327 0.131 1 0.065 0.137 0.202 0.338 0.353 o. 163 5 o. 147 0.183 0.259 0.584 0.473 0.232 10 0.220 0.209 0.303 0.655 0.545 0.274 25 0.271 0.234 0.340 0.697 0.595 0.294 50 (Median) 0.309 0.271 0.375 0.750 0.733 0.362 75 0.338 0.310 0.402 0.810 0.873 0.438 90 0.365 0.357 0.424 0.855 1 .072 0.535 95 0.379 0. 371 0.432 0.868 1.225 0.617 99 0.409 0.426 0.484 0.940 1.570 0.958 100 (Max.) 0.447 0.517 0.810 1.670 1. 762 1.058 (a) 25 18 16 20 ,. " "' 15 0 12 >- 0 c c " I 0 ~ " 10 "- tO 0> :l c:r " 8 ~ ..._ nTh I n 0 0.0 0.1 0.2 0.3 0.4 0.5 T r arrs formed Thinning Slope Pe1t (d) 35,--------------, " "' 30 25 0 20 c " 0 ~ 15 " "- 10 6 >- 0 r:: .. :l c:r ., 4 ~ ..._ ~....-4-in~n~n.....,....--+o 0.00 0.25 0.50 0.75 1.00 1.25 1.50 BSLA-weight Allometric Power 4.1? (b) " "' 0 c .. 0 ~ "' "- - 0 c "' :l c:r 15 "' 4 ~ ..._ ,- 10 ,- I h 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Height-weight Allometric Power ??? 35 30 25 >- 20 0 r:: "' :l 1 c:r 15 .. ~ ..._ 10 +---~r-~~~4-~~--+o 0.0 0.5 1.0 1.5 2.0 Heigh 1-DBH All ome tri c Power +.o (cl "' "' 0 c "' 0 ~ .. "- 30 25 20 15 10 1 +---~--~~~~~--~0 0.0 0.1 0.2 0.3 0.4 o.s >- 0 c ., :l c:r " ~ ..._ DBH-weight Allometric Power ~o. (f) 25 " "' 0 c "' 0 ~ ., "- 70 20 60 50 >- 0 15 r:: 40 ~ c:r ., 10 ~ 30 ..._ 20 10 ~~ ~m~.n ""~o 0.00 0.25 0.50 0. 75 1.00 1.25 1.50 Height-BSLA Allometric Power 4;.8 Figure 6.1. Histograms of allometric powers of the experimental and field data. (a) through (f) show, respectively, the statistical distributions of th~ tra~sformed !hinning slope, Pest' and the allometric powers $hw' ~OW' ~BW' ~hOt and ~hB? ..... w CX) (a} 20,---------------,- "' Ol 0 15 ~ 10 0 .... "' 70 (b) 60 50 >- 40 g "' 0> 0 ., " ~ 0" "' "' 30 " .... .... 18 (c) 45 140 16 40 50 14 35 120 12 40 30 100 >- ., >-0> " 0 " 10 c 25 c 30 ~ c 80 "' :J v " 0" " "' .... 20 "' .... .... []._ ..... "' "' 60 [l_ ..... a.. "- 20 10 0 0.0 0.1 0.2 0.3 0.4 0.5 Transformed Thinning Slope Pest (d) 25,---------------, 20 "' ~ 15 c "' " .... "' 10 a.. 60 40 20 n rl +-~~Juul~n~~~~4o 0.0 0.5 . 1.0 1.5 2.0 >- " c "' :J 0" "' .... ..... BSLA-weight Allometric Power ~ ?? (e) "' 0> 0 ~ "' " .... " [l_ 20 lJl 10 A fTh 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Height-weight All ome tri c Power ;nw 20 15 10 60 50 40 >? " c ., :J 30 g- 20 -+----+-l....._._._~lrh~-fTl-+-+-=--+:0 0.0 0.5 1.0 1.5 2.0 ... ..... Height-DBH Allometric Power ?,0 15 40 10 20 sf h 0 0.0 0.2 0.4 0.6 0.8 1.0 DBH-weight Allometric Power ?ow (f) 25 20 ., >-0> 15 0 0 c c ., ., 3 :J " 0" .... ., ., 10 .... [l_ "- 2 ~1--1-+-l---~ --,--.-.-+a 0.0 0.2 0.4 0.6 0.8 1.0 Height-BSLA Allometric Power~"" Figure 6.2. Histograms of allometric powers of the forestry yield table data. (a) through (f) show, respectively, the statistical distributions of the t~ansf~rmed thiQning slope, Pest? and the allometric powers ~hw? ~Ow? ~Bw? ~hD? and ~hB? w 1.0 Table 6.3. Correlations and Regressions Relating Transformed Thinninq Slooe to Allometric Powers. Linear Regression Equation Spearmann Pearson f\ 110- Correlation Correlation Slope Intercept metric 1\ r2 1\ Power n rs p r p 8 95% CI '). Q5% C T Exoerirnental and Field Data ----- 1\ * * ~hw 28 -0.51 0.0056 -0.55 0.0026 0.30 -0.71 [ -1.14,-0.26] 0.50 [ 0.36, 0.64] ~ .. 7 -0.57 0. 18 -0.63 0.13 0.40 -0.84 [-2.02, 0.34] 0.53 [ 0. 13, 0.94] /'yw , I) 6 -0.14 0.79 0.15 0.78 0.02 0.06 [-0.45, 0.56] 0. 1 R ~-0.36, 0.72] /\n * * wr18 24 -0.46 0.024 -0.44 0.032 0.19 -0.35 [-0.67,-0.03] 0.41 [ 0.28, 0.55] ~ 0 Forestry Yield Table Data ------ 1\ * * --' 1.2 142 to predict pest: 21% to 50% of the variance in pest was explained. ~ ~ Even ~Ow and ~Bw' which were not expected to relate well with Pest' were actually significantly correlated to Pest and explained 23% and 35%, respectively, of its variance. The relationship of Pest to ~hw explained 21% of the variance in Pest' and the regression line Pest = -0.54 ~hw + 0.445 was close to the expected equation p = -0.5 ~hw + 0.5 and did not differ significantly in slope from this line. The regression intercept of -0.445 was significantly less than expected, but the difference was small: the regression intercept was 11% below the expected 0.5. The position of the observed relationship below the expected one in the Pest-~hw plane is partly due to the invalid but necessary assumption that ~dw = 0. Figure 6.4 shows the relationships of Pest with $hw' ~hD' and ~hB" Small group sizes hindered the comparisons among the six groups of the EFD {herbaceous monocots, herbaceous dicots, temperate angiosperm trees, temperate gymnosperm trees, Eucalypts, and tropical angiosperm trees--see Table 6.4). Height was the most commonly measured plant dimension, yet only 28 of the 75 data sets that showed a significant log B-log N relationship also had height data and showed a significant log h-log w relationship. Group sizes were as low as one or two observations, so the absence of differences in $hw among the six groups is inconclusive. A significant difference among the four tree groups was observed for A ~ ~ ~hB' and tne mean values of S and ~hB for three groups were ranked in accordance with the allometric hypothesis: more (a) 0.7,-----------------, . . 0.. Q) 0.6 0.. 0 (/) 0.5 Ol c c 0.4 c .c I- 0.3 -o Q) ~ 0.2 0 -111 c 0.1 0 .... 1- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Height-weight Allometric Power ;hw (b) : 0.5 0.. Q) 0.. ~ 0.4 (/) Ol c ?;: 0.3 c .c 1- -o 0.2 Q) E .... 0 ";; 0.1 c 0 ... 1- 0.0+---~-~---,------~ 0 1 2 Height-DBH Allometric Power ~ho (c) ; o.5 0.. Ol c ?c a.3 c .c I- -o 0.2 Q) E .... 0 - 0.1 111 c 0 ... 1- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Heigh t-BS.LA All ome tri c Power ~hB Figure 6.4. Observed relationships between transformed thinning slope and allometric powers in the fqrestry yield table data. (a) shows the transformed thinning slope, Pest' plotted against sw ( 8) (12) (4) (4) P=0.86 P=0.70 P=0.58 P=0.54 Table 6.4. (continued) Group Meansa Tests for Differences Among Groups Herbs Trees All Groupsb Allo- metric Mono- Di- Temperate Temperate Tropical Kruskal- Power cots. cots. Angio. Gymno. Euca l . Angio. ANOVAc Wall isd A 1.56( 1,4) 1.19{1) ............., (I) 4 (I) 0 E 0 m 3 Q 0> 0 _J 2 ' ... ,, . .. ???:: ... ', . . . .. . .. ?? .. ',, .. ?. . . .?. :? ... ? ?????? ... ',, rll ? ?.: : ? ? . ' . . . . ... : .. : .. ??? ... . .. ,., . . . ?.,. ... ', .. ?. . . .. \ . ' . . .. .? .. . .',...._: . . . ??~ ....?.. ?. . . ',,,. . . ? .. ?.?. ???? ... . . . . ', . . ?. . .? ?? .. . ?. . . .. ',, . . .. .. . ?. . ' . . . .. ?. . ?. . . . ',,..... .. ?.? . ? ... ???... . '~: - .... ??' ?. . ?????? ... ??.. . ' . :. ~-.... .. : .. '~ . . ._ ?.. ..: .????~ ..?. . . ',~ . . ?????... - ' ' . : .. ? . ?? .. ~. ??.. . . . ~ .... ~. . .. ?. ??.... . ... ?. )'....... . .. : ?? ..... ?.. . . ', . . '\ ... . . ?.- ?.. .? .? ..... ', , .. \ .. ???... . . ...,, . . ....... ?????? .. . "?. .. ', ... . ?. . . . ?. ??... . ... '?.'", . . .. ?. .? ? ..... ~ . ?-.~~ ?~ ... ?.. . ?. . . ',~. . .. . ? ... ' . : .. ? .. . > \:;.:: ?=?: . ::--<-_.. :, : . ??~-~ .?; .. : :~ .. . ' ??... .. :.. . ',, ?. . ' ?. . . ' ???? .. :. ..? ',,, ?. . . ??;~ .. -2 0 2 4 Log 10 Density (plants/m 2) Figure 7. 1. Monte Carlo analysis of the improved model for the overall size-density relationship. The data points represent 500 hypothetical plant stands characterized by parameter values chosen from uniform random distributions (Table 7.1}. The solid line is the overall relationship log B = -0.53 log N + 4.28 (r2 = 0.96, P < 0.0001 95% CI for slope= [-0.56,-0.49]} fit by regressing log N against log w and transforming the resulting equation into an expression for log B as a function of log N. The dotted lines with intercepts of 3.33 and 5.10 are parallel to this line and enclose 90% of the data points. The dashed line is the relationship log B = -0.49 + 3.99 fitted to data from 65 stands of 29 plant species by Gorham (1979). 168 model, since other geometric solids that might represent real VO!s (cones, pyramids, spheroids, rectangular solids, hexagonal solids, and any general prismatoids) show the same power relationships of volume and base area to a linear dimension of the base. The variation around the overall trend is related to the ranges of possible values for plant shape and the density of biomass in occupied space. The more complex model yields similar predictions, thus demonstrating that the heuristic results are not artifacts of assuming that all plants in a given stand are the same size and shape. Both models can be used to understand why the overall relationship obtains and both show that the origin of this relationship is different from the allometric factors that determine the slopes of single species thinning lines (Chapter 6). The multi-species plots of Gorham {1979) and White (1980) address a fundamentally different question from an analysis of the self-thinning lines of individual populations. The former address the overall relationship determined by simple geometry while the latter addresses the time dynamics of space occupation by a single growing population. Because different phenomena are being examined, the existence of the overall relationship does not provide evidence for or against the self-thinning rule. It certainly does not show that self-thinning lines are insensitive to plant geometry. Instead, geometric differences are reflected by variations around the overall relationship. This interpretation would still admit the hypothesis that the self-thinning trajectories of individual stands all approximate the 169 line log w = -3/2 log N + 4 and so coincide with the overall relationship despite the differences in causality. This might seem justified on examining plots like that of White (1980) (Figure 7.2a), where thinning lines from many species seem to lie along a single line. However, in addressing real data, the log w-log N diagram gives a distorted view in which the appearance of linearity and high correlation is artificially enhanced while ~ariations among the data are hidden (Chapter 4). When the thinning lines are examined in an unbiased log B-log N diagram (Figure 7.2b) an overall relationship with a slope of -1/2 is present, but individual self-thinning lines vary widely in both slope and position around that trend. The same type of distortion is evident in Gorham's analysis. The log w-log N plot (Figure 7.3a) shows the fitted line log w = -1.49 log N + 3.99 and dotted lines enclosing 75% of the 65 stands examined (as in Gorham 1979). This plot appears to admit little potential variation from the overall trend and seems to justify the conclusion self-thinning lines must fall in a narrow band paralleling the overall relationship. However, the log B-log N plot of the same data (Figure 7.3b) with the dotted lines enclosing 90% of the observed stands shows that individual self-thinning lines can vary widely in slope and position from the overall trend yet still fall within the 90% limits. Thus, the overall relationship is consistent with different thinning slopes and intercepts for particular populations. In fact, the hypothesis that all self-thinning lines closely approximate log B = -l/2 log N + 4 has already been disproven (Chapter 5). (a) 6 ,........ 0'> '--" -.I: 0'> 4 Q) 3: Q) 0'> 2 0 !... Q) > <( 0 52 0'> 0 _J -2 -2 (b) 0 2 4 Log 10 Density (pI ants/ m 2) Vl Vl 0 E 0 m 3 52 0'> 0 _J 2 -2 0 2 4 Log 10 Density (plants/m 2) Figure 7.2. Previous analysis of the overall size-density relationship among thinning lines. (a) shows 65 previously cited self-thinning lines drawn by applying a reported thinning slope and intercepts (Table A.5) over the range of log N values covered by the thinning line (Table A.3). A similar plot of 31 thinning lines was constructed by White (1980). The dotted line is the overall relationship log w = -1.49 log N + 3.99 (Gorham 1979). (b) shows the same 65 self-thinning lines and Gorham's line replotted in the log B-log N plane. (a) 6 (b) 0 o?? .. 0 5.0 ,........, 0) 0 ,. o?? .. ,........, ......__, "?.9 N +- 0 E 0 0 ..r::: '-..... 4.5 0) 4 0) ......__, Q) . ., 3: " (/) 't~ .. (/) 4.0 ?q, 0 +- E 0 0 ?o 0 d?. ..r::: 2 m aD (/) ci'-. 3.5 D? ???a 0. +- jj 0 0 Q) ????? ... otb'?IOI 0 0) ?0 0 d' 0 fij 0 ?.'I) a??. .~~ 0 0 0 0 0 ?o.. 51 0 <( ?. 0 0) "' 0 51 0 _J 2.5 0 0) 0 0 _J c;. -2 -1 0 1 2 3 4 -1 0 1 2 3 Log 10 Density (plants/m 2) Log 10 Dens ft y (pI ant s/m 2) Figure 7.3. Gorham?s analysis of the overall size-density relationship among stands of different species. (a) shows data for 65 fully-stocked stands of species as analyzed by Gorham (1979). The solid line is log i = -1.49 log+ 3.99 and the dotted lines enclose 75% of the observations. In (b) the same data and fitted line are shown in a log B-log N plot, along with dotted lines enclosing 90% of the plant stands. -.....! 0 4 172 Further verification of the models for the overall relationship can be obtained ny comparing the predicted limits of variation around the relationship to the observed limits. This can be done by calculating a value K for a sample of real plant stands using K = B IN (7.14) which can be interpreted as the intercept at log N = 0 of the line log B = -1/2 log N +log K passing through the point (log N,log B). Log K is in some ways similar to log K, the intercept of a self-thinning line. Both K and K are constants of power equations and both are related to plant shape and the density of biomass in occupied space. However, the two constants are associated with fundamentally different phenomena. K relates to the time-dynamic self-thinning line determined by the particular allometry of a given stand while K relates to the overall relationship between size and density among all plant stands. Different symbols are used here to preserve this important distinction. Values of K were calculated here for the 1033 individual stands used to construct 75 self-thinning lines from experimental and field data (EFD) and the 3330 stands used to estimate 351 self-thinning lines for forestry yield table data (FYD). Chapter 5 details the data sources and self-thinning analyses. Table 7.3 presents summary statistics for calculated values of K and log K for both the EFD and the FYD while Figure 7.4 gives histograms for the distributions of log K. The range of log K ~? 173 Table 7. 3. Statistical Distributions of K and log K. and Field Dataa Forestry Yield Experimental Table data Shoot biomass Tot a 1 biomass All data sets Bole Biomass Statistic K log K K log K K log K K log K n 700 333 1033 3330 Mean 21830 4.01 32600 4.14 25300 4.05 7746 3.82 Std. dev. 43350 0.47 70440 0.46 53800 0.47 4196 0.27 cv t %) 199 12 216 11 213 12 54 7 Std. err. 1638 0.02 3860 0.03 1674 0.01 72.7 0.005 Skewness 5.00 0.78 3.92 1.37 4.73 0.93 0.87 -0.77 Kurtosis 30. 1 1.13 15.8 1. 79 25.1 1.40 1.60 0.80 f{ange 385300 3.49 479900 2.58 481000 3.58 30110 2.00 Percentiles 0 (Min.) 125 2.10 1257 3.10 125 2. 10 308 2.49 1 1010 3.00 3049 3.48 1671 3.22 1045 3.02 5 2815 3.45 4160 3.62 3074 3.49 2203 3.34 10 3461 3.54 4593 3.66 3992 3.60 2798 3.45 25 5115 3.71 7013 3.85 5437 3.74 4312 3.63 50 (Median) 7690 3.89 11030 4.04 8508 3.93 7325 3.86 75 17010 4.23 18290 4.26 17310 4.24 10460 4.02 90 52560 4.72 55890 4.75 52830 4.72 13460 4.13 95 82470 4.92 194500 5.29 91810 4.96 14820 4. 17 99 280100 5.45 394200 5.60 305700 5.49 18320 4.26 1 00 (Max.) 385500 5.59 481200 5.68 481200 5.68 30420 4.48 aThe three categories under this heading are based on the method of biomass measurement. "Shoot Biomass" includes thinning trajectories from biomass measurements of aboveground parts only. "Total Biomass" trajectories are based on biomass measurements that include aboveground and belowground parts. "All Data Sets" gives statistics for both of these groups combined. (a) 25 20 Q) 0> 15 0 c Q) (.) 1.... Q) 10 0... 5 0 2 - - ~ ,-,..--I 3 4 f- - 1- n-n 5 250 200 150 100 50 0 6 >- (.) c Q) :::J (J"" Q) 1.... lL.. (b) 16 14- 12- Q) 10 0> 0 -c 8 Q) (.) 1.... Q) 6-0... 4- 2- 0 2 ~ -n 3 4 5 Log 10 k 6 500 400 >? (.) 300 c Q) :::J (J"" Q) 1.... 200 lL.. 100 0 Figure 7.4. Histograms of log K. (a) shows the distribution of log K values for the 1033 data points of the experimental and field data, while (b) gives the distribution of log K for the 3330 data points of the forestry yield table data. 175 predicted by the heuristic model is 2.83 ~log K ~ 5.43 when the parameter ranges 600 ~ d ~40000 and 2 ~ T ~ 12 were used. Considering the extreme simplicity of the model and the crudity of the estimated parameter ranges, the agreement between the predicted range and the range observed for the EFD (2.10 ~log K ~ 5.68) is remarkable. Agreement is likewise good with the predicted range 3.01 ~log K ~ 5.47 of the more complex model. Values of log K for the FYD are consistently lower than the model predictions because many tree parts were not included in the wood volumes from which stand biomasses were estimated. The close agreement of predicted and observed K ranges, despite the crude model parameter estimates, supports the adequacy of the model as a representation of the overall relationship. The fact that simple models produce the overall relationship with very minimal information suggests that this relationship is, indeed, a symptom of a very simple geometric phenomenon and not an area of profound biological interest. Equation 7.4 can be transformed to log B = -112 log N + log(T d I in) and combined with equation 7.12 to give log K = log(T d I in) and K = T d I in. According to this equation, K can be interpreted as an composite measure of shape and density factors. The median values of log K near 4 observed in both the EFD and FYD corresponds to a value of 5600 glm3 for the product Td. The self-thinning rule has also been judged important in defining an ultimate thinning line which even constrains populations, such as clonal perennial herbs, that do not obey the self-thinning rule (Hutchings 1979}. Theories have been proposed to 176 explain the different patterns of biomass-density dynamics of clonal herbs (Hutchings and Barkham 1976, Hutchings 1979, Pitelka 1984), but all assume that plant populations are eventually constrained to lie below a line of slope -3/2 in the log w-log N plane. Hutchings (1979} proposed the line log w = -3/2 log N + 4.3 while White {as cited in Pitelka 1984) has suggested that few plant populations exceed?the size-density combinations defined by log w = -3/2 log N + 5. White (1981) stated that the constraint imposed by this ultimate thinning line demonstrates the importance of the self-thinning rule, even for species whose stand dynamics do not necessarily trace a line of slope -3/2 in the log w-log N plane. The ultimate thinning line is explained by the simple models, which predict that all stands must lie below the line defined by extreme values of certain model parameters. In the heuristic model, stand biomass can be increased by raising plant height so that taller, more massive columns of biomass rest on the same piece of ground. Alternatively, more biomass can be packed into each cubic meter of biological volume by increasing d. Structural requirements and genetic limitations must ultimately constrain how tall and thin plants can become and still remain upright in the face of gravity, wind, and rain. Energetic constraints must limit the density of biomass per unit of ground area. No photoautotroph can pack more grams of living biomass onto a given s~rface than can be maintained by the conversion of the radiant energy falling on that surface. Also, natural selection would not favor plants that waste energy maintaining biomass beyond the amount that would give adequate .. 177 structural support, maximal capture of resources, and maximum reproduction. The adequacy of the simple models for explaining the ultimate thinning line is supported by the good agreement between observed maximal values of log K and the maximal values predicted from crude guesses about the maximum values of T and d. The ninety-ninth percentile for log K in the EFD is approximately 5.0, supporting the assertion that most stands fall below the line log w = -3/2 log N + 5.0 (White, as cited in Pitelka 1984). However, the slope of this line is a consequence of simple geometry while its intercept is determined by energetic and structural constraints. A thorough exploration and quantitative evaluation of the general constraints on structure and biomass accumulation across the entire plant kingdom would be of great biological interest (and very difficult), but the existence of the ultimate thinning line of slope -3/2 is unsurprising. The term "ultimate thinning line" is also a misnomer. The constraint is not related to self-thinning dynamics and should hold regardless of how growth and mortality proceed in a given stand. In fact, a stand can approach this line by decreasing in average weight or actually increasing the density of individuals per unit area (Pitelka 1984), rather than by a mortality process as implied by the term "thinning." The overall relationship between size and density among stands of plants with different thinning behaviors also has significance for efforts to construct single species thinning lines. Since it is often difficult to obtain successive measurements of a given stand 178 through time, thinning lines have been commonly estimated from single measurements of stands of different ages. If the stands are grown under equivalent conditions and actually follow the same thinning trajectory, this method will give good results. However, if the stands are following different thinning lines, possibly due to genetic variations in plant geometry or differences in illumination or site fertility, then the thinning analysis will tend to reveal the overall relationship rather that the particular thinning trajectory of any of the stands. For example, Mohler et al. (1978) presented thinning data for Prunus pensylvanica and reported a thinning slope of -1.46 from fitting log~ against log N by principal component analysis. However, their data were collected from four sites in New Hampshire that were up to 100 km apart (Marks 1974). Elevations at these sites ranged from 340 to 570 m on north, south, and southeast facing slopes. At some sites, Prunus pensylvanica shared dominance with an important codominant Populus tremuloides. It seems likely that the large differences in slope, aspect, and other factors among sites many kilometers apart might mean that the stands would in fact follow different thinning trajectories. The analysis of Mohler et al. (1978) which gave a slope near -3/2 may then have revealed the overall relationship among stands following different thinning trajectories rather that the particular time trajectory of one of the stands. 179 CHAPTER 8 CONCLUSION This study has examined the self-thinning rule from both theoretical and empirical perspectives and made major advances in explaining observed size-density relationships. These include: (1) developing and analyzing explanatory models to produce testable hypothesis about underlying causes, (2) considering some important but largely unrecognized difficulties in testing the self-thinning rule and suggesting remedies for some problems, (3) completing a major analysis of self-thinning data to quantify the extent of variation in thinning line slope and intercept, (4) relating this variation to variations in plant allometry to verify that allometric factors are important in positioning the thinning line, and (5) developing models to explain the overall size-density relationship and evaluate its relevance to the self-thinning rule. The results of this study provide a basic explanatory theory for the rule and clarify its scientific importance. Several hypotheses about the cause of the self-thinning rule were derived from two mathematical models: a spatially averaged two-equation model and a simulation model that details individual sizes, locations, and competitive interactions. These hypotheses included: (1) Self-thinning lines are linear because plant dimensions and plant size are related by power functions. (2) The slope of the thinning line is determined by the allometry between the area occupied by an individual and its weight (the allometric 180 hypothesis). (3) The intercept of the thinning line is complexly related to plant allometry, the density of biomass in occupied space, and the partitioning of contested areas among competing individuals. Although the allometric hypothesis is not new (Westoby 1976, Miyanishi et al. 1979), the present models fill a major gap by deriving the hypothesis from time dynamic models rather than from ~hoc geometric arguments. The dynamic models can explain aspects of the thinning rule that simple geometric arguments can not, such as the dual nature of the thinning line as a time trajectory and a constraint separating feasible biomass-density combinations from impossible ones (Hozumi 1977, White 1981) and the change in thinning slope under low illumination (Westoby 1977). The models developed here emphasize a simple biological interpretation rather than an exact mathematical solution and relate the slope and intercept of the thinning line to meaningful ecological parumeters rather than to arbitrary constants; therefore, the models are heuristically useful and yield hypotheses that can be tested with measurable biological data. Chapter 4 considered the difficulties of testing the self-thinning rule and charted an optimal course for this task. Many potentially serious problems, such as the need for~ posteriori data editing, do not presently have a good solution. Such problems have been previously discussed (Mohler et al. 1978, Hutchings and Budd l98lb), but their gravity has not been fully appreciated. These limitations have been explored here so that their implications for the acceptance of the rule can be evaluated. Other aspects of 181 the analysis do have an optimal solution and much of the data supporting the self-thinning rule has not been analyzed in the best possible way. The problems include inattention to contradictory data, an invalid data transformation, inappropriate fitting techniques, and lack of hypotheses testing. In view of these problems, the results and interpretations of many studies are questionable. Chapter 5 presented the most exhaustive analysis of the self-thinning rule to date. Biomass and density data from previous self-thinning studies were reanalyzed to ensure that the selections of data, curve-fitting methods, and statistical interpretations were the most appropriate, and additional data not previously analyzed in a self-thinning context were included to broaden the body of test data. The data from previous studies did not provide strong support for the self-thinning rule. Many data sets that were claimed to corroborate the rule did not show any significant relationship between size and density, or gave a thinning slope different from the thinning rule prediction. The present analysis is also the most complete description of the variation in thinning slope and intercept, which are more variable than currently accepted. This greater variability was evidenced by by variations beyond currently accepted limits, by statistically significant differences in thinning slope and intercept among plant groups, and by significant correlations of slope and intercept with shade tolerance for forest trees. 182 The most important result of this study was the observation that variations in thinning slope are correlated with variations in plant allometry (Chapter 6). Such correlations were found in directly relating thinning slopes to allometric powers and in comparing among-group differences in thinning slope to among-group differences in allometric powers. This result supports the allometric hypothesis and verifies that the self-thinning rule is at least partly explainable by simple allometric arguments, despite previous failures to validate geometric models (Westoby 1976, Mohler et al. 1978, White 1981). Finally, the slope of the overall size-density relationship among stands of different species was shown to be a trivial consequence of the geometry of packing objects onto a surface and of the limitation of the plant shape and the density of biomass in occupied space to biologically reasonable values. As such, the overall relationship represents a phenomenon different from the individual self-thinning line and does not provide evidence for or against the self-thinning rule. The models suggest some additional analyses of self-thinning lines and tests of the hypotheses considered here. Some controlled experiments measuring thinning slope and several plant dimensions for species of differing allometric properties would be appropriate for verifying the evidence for the allometric hypothesis. It would also be instructive to gather data on plant allometry, density of biomass in occupied space, and the extent of overlap between 183 neighbors for several experimental populations to determine if these factors can explain the differences among thinning intercepts. The results of this study have important implications for the scientific importance of the self-thinning rule. This rule does not qualify as an ecological law. The accepted constancy of the proposed law~= K N- 312 (equivalently B = K N-l/2) is based on a body of data, analysis, and interpretation that is flawed in many ways. The slopes and intercepts of thinning lines are actually quite variable and can be explained by simple geometric models. The variability in the thinning slope is particularly important because S is actually an exponent in the power equation B = K NS so small differences in S represent major differences in the predicted course of growth and mortality. Since the slopes of thinning lines do not take the same constant value, they do not provide evidence of the operation of a single quantitative law. Sprugel (1984) has also recently suggested that the log w-log N thinning slope of -3/2 for trees is actually the exception rather than the rule. The recommendation of Pickard (1983) seems appropriate: It would now be profitable to proceed beyond the self-thinning rule to the analysis of deeper questions. 184 LITERATURE CITED 185 LITERATURE CITED Aikman, D. P., and A. R. Watkinson. 1980. A model for growth and self-thinning in even-aged monocultures of plants. Annals of Botany 45:419-427. Ashe, W. W. 1915. Loblolly or North Carolina pine. Bulletin 24, North Carolina Geological and Economic Survey, Raleigh, North Carolina, USA. Baker, F. S. 1925. Aspen in the central Rocky Mountain region. Department Bulletin 1291, United States Department of Agriculture, Washington D.C., USA. Baker, F. S. 1949. A revised tolerance table. Journal of Forestry 47:179-181. Bakuzis, E. V., and H. L. Hansen. 1965. Balsam fir Abies balsamea (Linnaeus) Miller: a monographic review. University of Minnesota Press, Minneapolis, Minnesota, USA. Barkham, J. P. 1978. Pedunculate oak woodland in a severe environment: Black Tor Copse, Dartmoor. Journal of Ecology 66:707-740. Barnes, G. H. 1962. Yield of even-aged stands of western hemlock. Technical Bulletin 1273, United States Department of Agriculture, Washington, D.C., USA. Bazzaz, F. A., and J. L. Harper. 1976. Relationship between plant weight and numbers in mixed populations of Sinapsis alba (L.) and Lepidium sativum (L.). Journal of Applied Eco1ogy-- 13:211-216. Beck, D. E. 1978. Growth and yield of white pine. Pages 72-90 in H. L. Williston and W. E. Balmer, editors. Proceedings: Symposium for the management of pines of the interior South. Technical Publication SA-TP2, United States Department of Agriculture, Washington, D.C., USA. Behre, C. E. 1928. Preliminary normal yield tables for second-growth western yellow pine in northeastern Idaho and adjacent areas. Journal of Agricultural Research 37:379-397. Bella, I. E. 1968. Jack pine yield tables for southeastern Manitoba. Publication 1207, Forestry Branch, Canadian Department of Fisheries and Forestry, Ottawa, Canada. 186 Bella, I. E. 1971. A new competition model for individual trees. Forest Science 17:364-372. Black, J. N. 1960. An assessment of the role of planting density in competition between red clover (Trifolium pratense L.) and lucerne (Medicago sativa L.) in the early vegetative stage. Oikos 11:26-42. Boisen, A. T., and J. A. Newlin. 1910. The commercial hickories. Forest Service Bulletin 80, United States Department of Agriculture, Washington D.C., USA. Boudoux, M. 1978. Empirical yield tables for jack pine. Forestry Chronicle 54:216-219. Bruce, D. 1923. Preliminary yield tables for second-growth redwood. Bulletin 361, University of California Agriculture Experiment Station, Berkely, California, USA. Bruce, D., and F. X. Schumacher. 1950. Forest mensuration. McGraw-Hill Book Co., New York, New York, USA. Cannell, M. G. R. 1982. World forest biomass and primary production data. Academic Press, London, England. Cary, N. L. 1922. Sitka spruce: Its uses, growth, and management. Bulletin 1060, United States Department of Agriculture, Washington D.C., USA. Charles-Edwards, D. A. 1984. On the ordered development of plants 2. Self-thinning in plant communities. Annals of Botany 53:709-714. Christensen, N. L., and R. K. Peet. 1982. Secondary forest succession on the North Carolina Piedmont. Pages 230-345 in D. C. West, H. H. Shugart, and D. B. Botkin, editors. Forest succession: Concepts and application. Springer Verlag, New York, New York, USA. Clark, P. J., and F. C. Evans. 1954. Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 35:445-453. Curtis, R. 0. 1971. A tree-area power function and related stand density measures for Douglas-fir. Forest Science 17:146-159. Dirzo, R., and J. L. Harper. 1980. Experimental studies on slug-plant interactions II. The effect of grazing by slugs on high density monoculturies of Capsella bursa-pastoris and Poa annua. Journal of Ecology 68:999-1011. ? 187 Drew, T. J., and J. W. Flewelling. 1977. Some recent theories of yield-density relationships and their application to Monterrey pine plantations. Forest Science 23:517-534 . Ernst, W. H ?? 1979. Population biology of Allium ursinum in Northern Germany. Journal of Ecology 67:347-362. Eyre, F. H. 1944. Management of jack pine stands in the Lake states. Technical Bulletin 863, United States Department of Agriculture, Washington D.C., USA. Eyre, F. H., and P. Zehngraff. 1948. Red pine management in Minnesota. Circular 778, United States Department of Agriculture, Washington, D.C., USA. Foiles, M. W. 1956. Effects of thinning on a 55-year old western white pine stand. Journal of Forestry 54:130-132. Forbes, R. D. 1961. Forestry handbook. The Ronald Press Co., New York, New York, USA. Ford, E. D. 1975. Competition and stand structure in some even-aged plant monocultures. Journal of Ecology 63:311-333. Forsythe, G. E., M.A. Malcolm, and C. B. Mohler. 1977. Computer methods for mathematical computations. Prentice Hall, Inc., Englewood Cliffs, New Jersey, USA. Fox, G. D., and G. W. Kruse. 1939. A yield table for well-stocked stands of black spruce in northwestern Minnesota. Journal of Forestry 37:565-567. Frothingham, E. H. 1912. Second-growth hardwood forests in Connecticut. Forest Service Bulletin 96, United States Department of Agriculture Washington, D.C., USA. Frothingham, E. H. 1931. Timber growing and logging practice in the Southern Appalachian region. Technical Bulletin 250, United States Department of Agriculture, Washington D.C., USA. Furnas, R. E. 1981. A resource theory of self-thinning in plant populations. Ph.D. dissertation. Cornell University. Ithaca, New York, USA. Gevorkiantz, S. R., and W. A. Duerr. 1937. northern hardwoods in the Lake states. 35:340-343. A yield table for Journal of Forestry 188 Gevorkiantz, S. R., and W. A. Duerr. 1939. Volume and yield of northern white cedar in the lake states: A progress report. Lake States Forest Experiment Station, Saint Paul, Minnesota, USA. Gold, H. J. 1977. Mathematical modeling of biological systems--an introductory guidebook. John Wiley and Sons, New York, New York, USA. Gorham, E. 1979. Shoot height, weight and standing crop in relation to density in monospecific plant stands. Nature 279:148-150. Haig, I. T. 1932. Second-growth yield, stand, and volume tables for the eastern white pine type. Technical Bulletin 323, United States Department of Agriculture, Washington D.C, USA. Hara, T. 1984. Modelling the time course of self-thinning in crowded plant populations. Annals of Botany 53:181-188. Harlow, W. M., E. S. Harrar, and F. M. White. 1978. Textbook of dendrology. McGraw-Hill Book Co., New York, New York, USA. Harper, J. L. 1977. Population biology of plants. Academic Press, New York, New York, USA. Harper, J. L., and J. White. 1971. The dynamics of plant populations. Pages 41-63 in den Boer, P. J., and G. R. Gradwell editors. Proceedings of the advanced study institute on dynamics of numbers in populations. Center for Agricultural Publishing. Oesterbeek, The Netherlands. Hatcher, R. J. 1963. A study of black spruce forests in northern Quebec. Publication 1018, Canadian Department of Forestry, Ottawa, Canada. Hickman, J. C. 1979. The basic biology of plant numbers. Pages 232-263 in 0. T. Solbrig, S. Jain, G. B. Johnson, and P. H. Raven, editors. Topics in plant population biology, Columbia University Press, New York, New York, USA. Hillis, W. E., and A. G. Brown. 1978. Eucalypts for wood production. Commonwealth Scientific and Research Organization, Adelaide, Australia. Hindmarsh, A. C. 1980. LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM (Association for Computing Machinery)-Signum Newsletter 15:10-11. r 189 Hirai, T., and M. Mansi. 1966. Dry matter economy of Helianthus annuus communities growing at varying densities and light intensities. Journal of the Faculty of Science of Tokyo University 9:241-285. Hozumi, K. 1977. Ecological and mathematical considerations on self-thinning in even-aged pure stands I. Mean plant weight-density trajectory during course of self-thinning. Botanical Magazine, Tokyo 90:165-179. Hozumi, K. 1980. Ecological and mathematical considerations on self-thinning in even-aged pure stands II. Growth analysis of self-thinning. Botanical Magazine, Tokyo 93:146-166. Hozumi, K., and K. Shinozaki. 1970. Studies on the frequency distribution of the weight of individual trees in a forest stand II. Exponential distribution. Japanese Journal of Ecology 20:1-9. Hutchings, M. J. 1975. Some statistical problems associated with determinations of population parameters for herbaceous plants in the field. New Phytologist 74:349-363. Hutchings, M. J. 1979. Weight-density relationships in ramet populations of clonal perennial herbs, with special reference to the -3/2 power law. Journal of Ecology 67:21-33. Hutchings, M. J. 1983. Ecology's law in search of a theory. New Scientist 98:765-767. Hutchings, M. J., and J. P. Barkham. 1976. An investigation of shoot interactions in Mercurialis perennis, a rhizomatous perennial herb. Journal of Ecology 64:723-743. Hutchings, M. J., and C. S. Budd. 198la. Plant competition and its course through time. Bioscience 31:640-645. Hutchings, M. J., and C. S. Budd, l98lb. Plant self-thinning and leaf area dynamics in experimental and natural monocultures. Oikos 36:319-325. Jacobs, M. R. 1979. Eucalypts for planting. FAO Forestry Series 11. Food and Agriculture Organization of the United Nations, Forestry Department, Rome, Italy. Jeffers, J. N. R. 1956. The yield of Hazel Coppice. Pages 12-30 in Utilization of Hazel Coppice. Bulletin 27, Forestry Commission, London, England. Johnson, P. L., editor. 1977. A ecosystem paradigm for ecology. Oak Ridge Associated Universities, Oak Ridge, Tennessee, USA. 190 Johnson, N. L., and S. Kotz. 1970. Continuous univariate distributions-1. Houghton Mifflin Co., Boston, Massachusetts, USA. Jolicoeur, P. 1973. Imaginary confidence limits of the slope of the major axis of a bivariate normal distribution: A sampling experiment. Journal of the American Statistical Association 68:866-871. Jolicoeur, P. 1975. Linear regressions in fishery research: Some comments. Journal of the Fisheries Research Board of Canada 32:1491-1494. Jolicoeur, P., and A. A. Heusner. 1971. The allomery equation in the analysis of the standard oxygen consumption and body weight of the white rat. Biometrics 27:841-855. Jones, R. M. 1982. -3/2 power law. Bioscience 32:6. Kays, S., and J. L. Harper. 1974. The regulation of plant and tiller density in a grass sward. Journal of Ecology 62:97-105. Khil 1 mi, G. F. 1957. Theoretical forest biogeophysics. Israel Program for Scientific Translation, Ltd., Jerusalem, Israel. Kittredge, J., and S. R. Gevorkiantz. 1929. Forest possibilities of aspen lands in the Lake states. Technical Bulletin 60, University of Minnesota Agricultural Experimental Station, Saint Paul, Minnesota, USA. Koristan, C. F. 1931. Southern white cedar. Technical Bulletin 251, United States Department of Agriculture, Washington, D.C., USA. Koyama, H., and T. Kira. 1956. Frequency distribution of individual plant weight as affected by the interaction between plants (Intraspecific competition among higher plants VII). Journal of the Institute of Polytechnics, Osaka City University, Series D 7:73-94. Lindquist, J. L., and M. N. Palley. 1963. Empirical yield tables for young-growth redwood. Bulletin 796, California Agricultural Experiment Station, University of California, Berkeley, California, USA. Lonsdale, W. M., and A. R. Watkinson. 1982. Light and self-thinning. New Phytologist 90:431-445. Lonsdale, W. M., and A. R. Watkinson. 1983a. Plant geometry and self-thinning. Journal of Ecology 71:285-297. 191 Lonsdale, W. M., and A. R. Watkinson. 1983b. Tiller dynamics and self-thinning in grassland habitats. Oeco1ogia 60:390-395. Madansky, A. 1959. The fitting of straight lines when both variables are subject to error. Journal of the American Statistical Association 54:173-205. Malmberg, C., and H. Smith. 1982. Relationship between plant weight and density in mixed population of Medicago sativa and Trifolium pratense. Oikos 38:365-368. Marks, P. L. 1974. The role of pin cherry (Prunus pensylvanica L.) in the maintenance of stability in northern hardwood ecosystems. Ecological Monographs 44:73-88. Martin, G. L., A. R. Ek, and R. A. Monserud. 1977. Control of plot edge bias in forest stand growth simulation models. Canadian Journal of Forest Research 7:100-105. Marty, R. 1965. The mensurational characteristics of eastern white pine. Forest Service Research Note NE-40, Northeastern Forest Experiment Station, Upper Darby, Pennsylvania, USA. Mattoon, W. R. 1915. Shortleaf pine: Its economic importance and forest management. Bulletin 308, United States Department of Agriculture, Washington, D.C., USA. McArdle, R. E. 1930. The yield of douglas fir in the Pacific Northwest. Bulletin 201, United States Department of Agriculture, Washington, D.C., USA. McArdle, R. E., W. H. Meyer, and D. Bruce. 1949. The yield of douglas fir in the Pacific Northwest. Technical Bulletin 201, 1949 Revision, United States Department of Agriculture, Washington, D.C., USA. McCarthy, E. F. 1933. Yellow poplar characteristics, growth, and management. Technical Bulletin 356, United States Department of Agriculture Washington, D.C., USA. Mcintosh, R. P. 1980. The background and some current problems of theoretical ecology. Synthese 43:195-255. Meyer, W. H. 1929. Yields of second growth spruce and fir in the Northeast. Technical Bulletin 142, United States Department of Agriculture, Washington, D.C., USA. Meyer, W. H. 1937. Yield of even-aged stands of sitka spruce and western hemlock. Technical Bulletin 544, United States Department of Agriculture, Washington, D.C., USA. 192 Meyer, W. H. 1938. Yield of even-aged stands of ponderosa pine. Technical Bulletin 630, United States Department of Agriculture, Washington, D.C., USA. Miyanishi, K., A. R. Hoy, and P. B. Cavers. 1979. A generalized law of self-thinning in plant populations. Journal of Theoretical Biology 78:439-442. Mohler, C. L., P. L. Marks, and D. G. Sprugel. 1978. Stand structure and allometry of trees during self-thinning of pure stands. Journal of Ecology 66:599-614. Moran, P. A. P. 1971. Estimating structural and functional relationships. Journal of Multivariate Analysis 1:232-255. Obeid, M., D. Machin, and J. L. Harper. 1967. Influence of density on plant to plant variation in fiber flax, Linum usitatissimum L. Crop Science 7:471-473. o?Neill, R. V., and D. L. DeAngelis. 1981. Comparative productivity and biomass relations of forest ecosystems. Pages 411-449 in D. E. Reichle, editor. Dynamic properties of forest ecosystems. Cambridge University Press, Cambridge, England. Oshima, Y., M. Kimura, H. Iwaki, and S. Kuroiwa. and physiological studies on the vegetation Prelimina.ry survey of the vegetation of Mt. Botanical Magazine, Tokyo 71:289-301. 1958. Ecological of Mt. Shimagare I. Shimagare. Patton, R. T. 1922. Red oak and White oak: A Study of growth and yield. Harvard Forest Bulletin 4, Harvard University, Cambridge, Massachusetts, USA. Peattie, D. C. 1950. A natural history of trees of eastern and central North America. Houghton Mifflin Co., Boston, Massachusetts, USA. Peattie, D. C. 1953. A natural history of western trees. Houghton Mifflin Co., Boston, Massachusetts, USA. Peet, R. K., and N. L. Christensen. 1980. Succession: A population process. Vegetatio 43:131-140. Perry, D. A. 1984. A model of physiological and allometric factors in the self-thinning curve. Journal of Theoretical Biology 106:383-401. Pickard, W. F. 1983. Three interpretations of the self-thinning rule. Annals of Botany 51:749-757. .. 193 Pitelka, L. F. 1984. Application of the -3/2 power law to clonal herbs. American Naturalist 123:442-449. Pollard, D. F. W. 1971. Mortality and annual changes in distribution of above-ground biomass in an aspen sucker stand. Canadian Journal of Forest Research 1:262-266. Pollard, D. F. W. 1972. Above-ground dry matter production in three stands of trembling aspen. Canadian Journal of Forest Research 2:27-33. Popper, K. R. 1963. Conjectures and refutations: The growth of scientific knowledge. Harper and Row, New York, New York, USA. Puckridge, D. W., and C. M. Donald. 1967. Competition among wheat plants sown at a wide range of densities. Australian Journal of Agricultural Research 18:193-211. Rabinowitz, D. 1979. Bimodal distributions of seedling weight in relation to density of Festuca paradoxa Desv. Nature 277:297-298. Reineke, L. H. 1933. Perfecting a stand-density index for even-aged forests. Journal of Agricultural Research 46:627-638. Ricker, W. E. 1973. Linear regressions in fishery research. Journal of the Fisheries Research Board of Canada 30:409-434. Ricker, W. E. 1975. A note concerning Professor Jolicoeur's comments. Journal of the Fisheries Research Board of Canada 32:1494-1498. Riggs, D. S. 1963. The mathematical approach to physiological problems: A critical primer. M.I.T. Press, Cambridge, Massachusetts, USA. Ross, M. A., and J. L. Harper. 1972. Occupation of biological space during seedling establishment. Journal of Ecology 60:77-88. Schaffer, W. M., and E. G. Leigh. 1976. The prospective role of mathematical theory in plant ecology. Systematic Botany 1 :209-232. Schlesinger, W. H. 1978. Community structure, dynamics and nutrient cycling in the Okefenokee cypress swamp forest. Ecological Monographs 48:43-65. Schlesinger, W. H., and D. S. Gill. 1978. Demographic studies of the chaparral shrub, Ceanothus megacarpus, in the Santa Ynez Mountains, California. Ecology 59:1256-1263. 194 Schnurr, G. L. 1937. Yield, stand, and volume tables for even-aged upland oak forests. Technical Bulletin 560, United States Department of Agriculture, Washington, D.C., USA. Schumacher, F. X. 1926. Yield, stand, and volume tables for white fir in the California pine region. Bulletin 407, University of California Agriculture Experiment Station, Berkeley, California, USA. Schumacher, F. X. 1930. Yield, stand, and volume tables for Douglas fir in California. Bulletin 491, University of California Agriculture Experiment Station Berkeley, California. Schumacher, F. X., and T. S. Coile. 1960. Growth and yield of natural stands of the southern pines. T. S. Coile, Inc., Durham, North Carolina, USA. Show, S. B. 1925. Yield capacities of the pure yellow pine type on the east slope of the Sierra Nevada Mountains in California. Journal of Agricultural Research 31:1121-1135. Smith, J. H. G. 1968. Growth and yield in red alder in British Columbia. Pages 273-286 in J. M. Trappe, J. F. Franklin, R. F. Tarrout, and G. M. Hansen, editors. Biology of alder: Proceedings of a Symposium held at Northwest Scientific Association, Fortieth Annual Meeting, Pullman, Washington, April 14-15, 1967. United States Forest Service Pacific Northwest Forest Experiment Station, Portland, Oregon, USA. Snedecor, G. W., and W. G. Cochran. 1956. Statistical methods. The Iowa State University Press, Ames, Iowa, USA. Sokal, R. R., and F. J. Rohlf. 1981. Biometry. W. H. Freeman and Co., San Francisco, California, USA. Sprent, P., and G. R. Dolby. 1980. The geometric mean functional relationship. Biometrics 36:547-550. Sprugel, D. G. 1984. Density, biomass, productivity, and nutrient-cycling changes during stand development in wave-regenerated balsam fir forests. Ecological Monographs 54:165-186. Spurr, S. H., L. J. Young, B. v. Barnes, and E. L. Hughes. 1957. Nine successive thinnings in a Michigan white pine plantation. Journal of Forestry 55:7-13. Steill, W. M. 1976. White spruce: Artificial regeneration in Canada. Canadian Forestry Service, Department of the Environment, Ottawa, Canada. , 195 Steill, W. M., and A. B. Berry. 1973. Yield of unthinned red pine plantations at the Petawawa forest experiment station. Publication 1320, Canadian Forestry Service, Ottawa, Canada. Sterrett, W. D. 1915. The ashes: Their characteristics and management. Bulletin 229, United States Department of Agriculture, Washington, D.C., USA. Tadaki, Y., and T. Shidei. 1959. Studies on the competition of forest trees II. The thinning experiment on a small model stand of sugi (Cryptomeria japonica) seedlings. Journal of the Japanese Forestry Society 41:341-349. Taylor, R. F. 1934. Yield of second-growth western hemlock-sitka spruce stands in Southwestern Alaska. Technical Bulletin 412, United States Department of Agriculture, Washington, D.C., USA. Tepper, H. B., and G. T. Bamford. 1960. Thinning sweetgum stands in New Jersey. Forest Research Note NE~95, United States Department of Agriculture Northeastern Forest Experiment Station, Upper Darby, Pennsylvania, USA. Tseplyaev, V.P. 1961. The forests of the USSR. Israel Program for Scientific Translation, Ltd., Jerusalem, Israel. United States Department of Agriculture. 1929. Volume, yield,. and stand tables for second-growth southern pines. Miscellaneous Publication 50. United States Department of Agriculture, Washington, D.C., USA. Vermont Agricultural Experiment Station. second growth hardwoods in Vermont. Publication 13, Vermont Agricultural Burlington, Vermont, USA. 1914. The management of Forest Service Experiment Station, Wahlenberg, W. G. 1946. Longleaf pine: Its use, ecology, regeneration, growth, and management. Charles Lathrop Park Forestry Foundation, Washington D.C., USA. Wahlenberg, W. G. 1955. Six thinnings in a 56-year-old pure white pine plantation at Biltmore. Journal of Forestry 53:331-339. Watkinson, A. R. 1980. Density-dependence in single-species populations of plants. Journal of Theoretical Biology 80:344-357. Watkinson, A. R. 1984. Yield-density relationships: the influence of resource availability on growth and self-thinning in populations of Vulpia fasciculata. Annals of Botany 53:469-482. 196 Watkinson, A. R., W. M. Lonsdale, and L. G. Firbank. 1983. A neighborhood approach to self-thinning. Oecologia 56:381-384. Westoby, M. 1976. Self-thinning in Trifolium subterraneum not affected by cultivar shape. Australian Journal of Ecology 1:245-247. Westoby, M. 1977. Self-thinning driven by leaf area not by weight. Nature 265:330-331. Westoby, M. 1981. The place of the self-thinning rule in population dynamics. American Naturalist 118:581-587. Westoby, M. and L. Brown. 1980. The effect of clipping on self-thinning in Trifolium pratense. Australian Journal of Ecology 5:407-409. Westoby, M., and J. Howell. 1981. Self-thinning: The effect of shading on glasshouse populations of silver beets (Beta vulgaris). Journal of Ecology 69:359-365. Westoby, M., and J. Howell. 1982. Self-thinning in Trifolium subterraneum populations transferred between full daylight and shade. Journal of Ecology 70:615-621. Weth~y, D. S. 1983. Intrapopulation variation in growth of sessile organisms: Natural populations of the intertidal barnacle Balanus balanoides. Oikos 40:14-23. White, J. 1977. Generalization of self-thinning of plant populations Comment on Westoby(l977). Nature 268:337. White, J. 1980. Demographic factors in populations of plants, Pages 21-48 in 0. T. Solbrig, editor. Demography and Evolution in Plant Popllfations. University of California Press, Berkeley, California, USA. White, J. 1981. The allometric interpretation of the self-thinning rule. Journal of Theoretical Biology 89:475-500. White, J., and J. L. Harper. 1970. Correlated changes in plant size and number in plant populations. Journal of Ecology 58:467-485. White, J., and S. J. Gould. 1965. Interpretation of the coefficient in allometric equations. American Naturalist 99:5-18. 197 Whittaker, R. H. and G. M. Woodwell. 1968. Dimension and production relations of trees and shrubs in the Brookhaven Forest, New York. Journal of Ecology 56:1-25. Williamson, A. W. 1913. Cottonwood in the Mississippi Valley. Bulletin 24, United States Department of Agriculture, Washington D.C., USA. Winters, R. K., and J. G. Osborne. 1935. Growth and yield of second-growth red gum in fully stocked stands on alluvial lands in the South. Occasional Paper 54, Southern Forest Experiment Station, New Orleans, Louisiana, USA. Yoda, K., T. Kira, H. Ogawa, and K. Hozumi. 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions. (Intraspecific competition among higher plants XI). Journal of the Institute of Polytechnics, Osaka City University, Series D 14:107-129. 198 APPENDIXES 199 APPENDIX A EXPERIMENTAL AND FIELD DATA Table A.l. Sources of Exoerimental and Field Data ld. Study Data Codea Species Groupb TypeC Typed Condit i onse Referencef l Prunus pensylvanica TTmA N A Marks 1974, Tabs. 2,3 5 Populus tremuloides TTmA N A Pollard 1971, Tab. l Pollard 1972, Tab. 2 Triticum9 HM F TO Puckridge and Donald 1967, Fig. l, Tab. 2; White and Harper 1970, Fig. 8 Pinus strobus TTmG p T Plot Spurr et al. 1957, Tabs. 1,2 10 Trifolium subterraneum. HD E T Westoby and Howell 1982, Tab. 14 L1qu1dambar styraciflua TTmA N T Tepper and Bamford 1960, Tab. 15 Erigeron canadensis HD N N Yoda et al. 1963, Fig. 14 16 Plantago asiatica HD N N Yoda et al. 1963, Fig. 13 17 Amaranthus retroflexus HD N N Yoda et al. 1963, Fig. 16 18 Ambrosia artimestifolia HD N N Yoda et al. 1963, Fiq. 15 19 A61es sachal1nens1s TTmG N N Yoda et al. 1963, Fig. 22 20 Chenopodium album HD N N Yoda et al. 1963, Fig. 17 21 Er1geron canadensis HD F T Yoda et al. 1963, Tabs. 2,3 22 Prunus pensylvanica TTmA N A Mohler et al. 197R 23 Abies balsamea TTmG N A Mohler et al. 1978 24 Ab1es sachal1nesis TTmG N N Hozumi and Shinozaki 1970, Tab. 2A 26 Betula TTmA N N Yoda et al. 1963, Fig. 23 N 27 Tr1folium praetense HD E T Westoby and Brown 1980, Fiq. 2 0 28 Trifol1um praetense HD E T Illumination Hutchings and Budd 198lb, Fig. 4 0 29 Fagopyrum esculentum HD F TO Yoda et al. 1963, Fio. 21 30 Trifolium praetense HD F TD Black 1960, Fig. 1, Tabs. 1-3 31 Medicago sativa HD F TD Black 1960, Fig. 1, Tabs. 1-3 32 Fagopyrum esculentum HD E TO Fertility Furnas 1981, Fig. 4.1 33 Beta vulgaris HD E TO Furnas 1981, Fig. 4.2 34 Beta vulgaris HO E T Furnas 1981, Fig. 4.5 35 Trifolium subterraneum HD E T Westoby 1976, Fig. 1 36 Ceanothus megacarpus s N A Schlesinger and Gill 1978, Fig. 3, Tab. 38 Lo li um perenne HM E TO Illumination Kays and Harper 1974, Fig. 4 39 Betula TTmA N N Site Hozumi and Shinozaki 1970, Tab. 2B,2C 40 Tagetes patula HD E TO Ford 1975, Fig. 12 41 Quercus robur TTmA N A Barkham 1978, Tab. 5 43 Beta vulga.rls HD E T Ill umi nation Westoby and Howell 1981, Fig. 1 44 Helianthus annuus HD F TO Experiment; Hiroi and Monsi 1966, Fiqs. 1,5,6, Tab. 2 Illumination 45 Picea mariana TTmG N A Hatcher 1963, Tab. 11 48 fo yl s ave 11 ana TTmA c A Jeffers 1956, Tabs. 16-18 49 agooyrum esculentum HD E TO Furnas 1981, Fig. 4.6 50 Brassica juncea HD F TO Illumination Furnas 1981, Fig. 4.10 51 Sinapsis alba and HD E T Bazzaz and Haroer 1976, Fiq. Leo1dium sativum 52 Cryptomeri a j apom ca TTmG F T Tadaki and Shidei 1959, Tabs. 1-3 53 Brassica napus and HD E T White and Haroer 1970, Fiq. 2 Raphanus sativus ? t Table A.l. (continued) ]d. Study Data Referencef Codea Species Groupb TypeC Typed Conditionse 54 Larix occidentalis and TTmG --rTnus monticola N T Foiles 1956, Tabs. 1,2 55 Trifolium Praetense HD E T Malmberg and Smith 19R2, F g. 4 56 Medicago sativa HD E T Malmberg and Smith 1982, F g. 4 57 Medicago sativa and HD E Tr1folium pratense T Malmberg and Smith 1982, F g. 3 58 Beta vulgaris and HD F --srassica JUncea T Furnas 1981, Figs. 4.8,4.9 80 Taxodium distichum TTmG N N Schlesi~ger 1978, Fig. 3C 81 Ab1es veitchil and TTmG N A Oshima et al. 1958, Tab. 2, Fig. 4 ~es mar1esii 82 Pinus taeda TTmG N AD Peet and Christensen 1980, Fiq. 3; Christensen and Peet 1982, Fig. 15.2 84 Pinus densiflora TTmG N N Yoda et al. 1963, Fiq. 24 86 Festuca pratensis HM E T Lonsdale and Watkinson 1983a, Figs. 1,2C 87 Agrostemma ~ HD E T Lonsdale and Watkinson 1983a, Figs. 1,2C 88 Chicorium endivium HD E T Lonsdale and Watkinson 1983a, Figs. 1,2C 89 Capsella bursa-pastoris HD E T Grazing level Dirzo and Haroer 1980, Fig. 3 90 Poa annua HM E T Grazing level Dirzo and Haroer 1980, Fig. 3 N 91 IoTi urn oerenne HM E TO Illumination Lonsdale and Watkinson 1982, Fig. 1 0 92 Lolium oerenne HM E T Illumination Lonsdale and Watkinson 1982, Fig. 5 __, 93 P1nus strobus TTmG p T Beck 1978, Tab. 2; ----- Wahlenberg 1955, Fig. 2 95 Eucalyptus obliqua TE p T Hillis and Brown 1978, Tab. 10.17 96 Eucalyptus pilularis TE N T Hillis and Brown 1978, Tab. 10.21 97 Eucaltotus regnans TE N T Hillis and Brown 1978, Tab. 10.24 98 Eucaltotus reglans TE p T Site Index Hillis and Brown 1978, Tab. 10.26 99 Eucalyptus deg upta TE p T Hillis and Brown 1978, Tab. 10.34 101 ~ucaltotus grand1s TE p A Cannell 1982, Page 11 102 Pinus taeda TTmG N A Cannell 1982, Page 320 103 ~agus sylvatica TTmA N A Cannell 1982, Page 31 104 ~ sylvatica TTmA N A Cannell 1982, Page 58 105 Fagus srlvatica TTmA N A Cannell 1982, Page 72 106 Acer sp1catum TTmA N A Cannell 1982, Page 33 109 M1xed hardwoods TTmA N A Cannell 1982, Page 34 E. Canada) 110 Pinus banksiana TTmG N A Cannell 1982, Page 44 lll Shorea robusta TTrA p A Cannell 1982, Page 79 112 C~clobalanops1s TTmA c myrs1naefol1a A Cannell 1982, Paqe 108 113 Tectona grand1s TTrA p A Cannell 1982, Page 83 114 Abies veitchii TTmG N A Cannell 1982, Page 129 115 ~l1a Jaoonica TTmA N D Cannell 1982, Page 9'! 116 Eucaltotus obliqua TE N A Cannell 1982, Page 14 117 Abies firma and TTmG N A Cannell 1982, Page 124 Tsuga sieboldii Table A.l. (continued) I d. Study Data Codea Species Groupb TypeC Typed Conditionse Referencef ll8 Eucalyptus tereticornis TE p A Cannell 1982, Page 78 ll9 Abies sachalinensis TTmG p A Cannell 1982, Page 126 120 Cryptomeria japon1ca TTmG p A Cannell 1982, Page 151 121 BetulaY TTmA N A Cannell 1982, Page 238 122 Cryptomeria japonica TTmG p A Cannell 1982, Page 146 123 Castanea sativa TTmA c A Cannell 1982, Page 23CJ 124 Ab1esY TTmG N A Cannell 1982, Page 133 125 P1nus sylvestris TTmG p A Cannell 1982, Page 243 126 Quercus pubescens TTmA N A Cannell 1982, Page 68 128 Pinus pum1la TTmG N A Cannell 1982, Page 176 130 Pinus nigra TTmG p A Cannell 1982, Page 243 131 Alnus rubra TTmA N A Cannell 1982, Page 252 132 CryptoiiieMa japonica TTmG p A Cannell 1982, Page 146 133 Populus deltoides TTmA N A Cannell 1982, Page 266 135 Pinus banksiana and TTmG N A Cannell 1982, Page 300 ~ed hardwoods 136 Picea abies TTmG p A Cannell 1982, Page 73 137 P1cea ab1es TTmG N A Cannell 1982, Page 361 138 Alnus incana and TTmA N -saT;~ A Cannell 1982, Page 251 aAn arbitrary number assigned to facilitate cross-referencing among data tables. bone of seven categories. HM = herbaceous monocots; HD = herbaceous dicots; S shrubs; TTmA = trees, temperate angiosperms; TTmG = trees, temperate gymnosperms; TTrA = trees, tropical angiosperms; and TE = trees of genus Eucalyptus. CLevel of experimental control. N =natural field populations, P =tree olantations, C = coppiced trees, F =outdoor experiments, and E =greenhouse or light chamber experiments. drndicates how observations of stands at different densities and biomasses were generated. T = repeated measurements of a single population (time series), A= observations of populations of different ages (age series), 0 =observations of populations started at different initial densities (density series), N = not specified. eNatural conditions or experimental treatments that may have affected thinning line slope or position, so that separate thinning lines were fitted to subsets of the data points from a single source. "Plot" and "site" both refer to different physical locations at which plant populations were observed. "Illumination" means that different populations received to different intensities of light. "Grazing level" and "fertility" likewise indicate that different populations were subjected to different levels of the indicated stress or resource. "Experiment" refers to different experiments that were considered separately in the reference. "Site index" is a measure of site quality used by foresters, here the height in meters of the largest trees when the stand is 20 years old. See the references given for further details. fTabular data were taken from the indicated table numbers. Figure numbers refer to graphs from which data were reconstructed. gGeneric name only indicates that the specific name was not given in the reference, or that more than one species from the genus were present. N 0 N .. Table A.2. Self-thinning Lines Fit to Exoerimental and Field Data No. of PCA Self-thinning Lin~e PointsC Slooe Jnterceot c~~~a Condition b 2 pd a %% Cl f a 95% CJ q nT r 0 est lA 4 3 0.9fl3 O.Ofl29 -0.429 3.77 0.350 1T 4 3 0.980 0.0911 -0.424 3.R4 0.351 SA 7 4 0.941 0.0300* -0.264 [-0.484, -0.065]* 3.65 [3.61, 3.67] 0.396 7A 33 5 0.985 O.OOOfl* -0.362 [ -0.448, -0.2RO]* 3.85 [3.60, 4.11 J 0.367 SA Lot 2B 9 7 0.986 <0.0001* -0.724 [-0.830, -0.62R]* 3. 78 [ 3. 73, 3.83] 0.290 8A Lot 2C 8 7 0.987 <0.0001* -1.116 [-1.278, -0.976]* 3.34 [3.2?, 3 .43] 0.236 lOA Full 1 ight 10 10 0.836 0.0002* -0.473 [-0.660, -0.310] 4.60 [ 3. 97. 5.33] 0.330 14A 6 4 0.071 0.0145* -0.667 [-1.117, -0.34fl] 3. 71 [3. 37. 3.96] 0.300 15T 47 31 0.930 <0.0001* -0.621 [-0.68R, -0.558]* 4. 36 [ 4. 19, 4 .55] 0.308 16T 38 3R 0.926 <0.0001* -0.472 [ -0.518, -0.428] 3.98 [3.83, 4. 13] 0.340 17T 21 17 0.898 <0.0001* -0.588 [ -0.703, -0.483] 4.06 [3.80, 4. 35] 0.315 18T 51 43 0.886 <0.0001* -0.539 [ -0.601, -0.479] 3.80 [3. 66, 3. 96] 0.325 19A 28 26 0.841 <0.0001* -0.649 [-0.776, -0.535]* 4.16 [ 4. 15, 4. 17] 0.303 20T 27 13 0.929 <0.0001* -0.405 [-0.4fl2, -0.332]* 5.15 [ 4. 87. 5 .45] 0.356 21T 30 13 0.987 <0.0001* -1.038 [-1.121, -0.962]* 5. 70 r5.43, 6.00] 0.?4S 22T 53 34 0.395 <0.0001* -0.410 [-0.611, -0.234] 3.92 [3 .82. 4.041 0.355 23T 29 23 0.654 <0.0001* -0.204 [ -0.273, -0.13R]* 3.88 [3.88, 3.89] 0.415 24T 9 5 0.839 0.0290* -0.465 [ -0.965, -0.104] 4.39 [4.07. 4.83] 0.341 N 26A 46 27 0.656 <0.0001* -1.033 [ -1.406, -0. 760]* 3.83 [ 3. 73, 3.89] 0.246 0 27A 5 2 1.000 -0.313 4.33 0.3Rl w 28A Low L. I. 10 7 0.394 0.1310 -0.851 4.62 0.270 ?8A Med. L. I. 14 R 0.137 0.3666 -1.192 6.53 0.228 28A High L. I. 14 7 O.ll61 0.0026* -0.627 [ -0.978, -0.360] 4.78 [3.95, 5.1l6] 0.307 29T 25 9 0.954 <0.0001* -0.690 [-0.836, -0.561]* 5.08 [4.68, 5. 55] 0.296 30A 22 9 0.137 0.3266 -0.608 4.89 0.311 31A 22 11 0.180 O.l93R -0.434 3.95 0.349 32T 20% N.C. 7 5 0.003 0.9289 -0.024 2.86 0.488 32T 100% N.C. 7 5 0.024 0.8027 -0.029 [-0.430, 0.362] 3.21 [1.88, 4.59] 0.48n 33T 10 10 0.592 0.0092* -1.335 [-3.355, -0.64R]* 6.3R [4.54, 11.80] 0.214 34T 10 10 0.645 0.0052* -2.304 [-5.47R, -1.348]* 9.93 [7 .OR, 19.38] 0.151 35A 22 13 0.716 0.0003* -0.622 [ -0.928, -0.382] 5.17 [ 4. 33, 6.23] 0.308 36A 11 10 0.226 0.1653 -0.188 [ -0.514, 0. 105] 3.64 [3.49, 3.RO] 0.421 38A 30% L.J. 20 7 0.054 0.6144 0.055 [-0.227, 0.347] 1. 98 [ 1.03, 2.89] 0.529 38A 70% L.J. 20 8 0.422 0.0814 -0.452 [ -1.429, o. 112] 4.16 [2.3fi, 7.28] 0.344 3qA 100% L. I. 20 8 0.549 0.035A* -0.324 [ -0.674, -0.034] 3. 79 (2.88, 4.R9] 0.378 39A Kit ami 6 5 0.752 0.0569 -0.429 [-1.118, 0.030] 4.85 [4.52, 5.07] 0.350 39A Shirogane 7 7 0.043 0.6554 0.051 [-0.247, 0.359] 4.69 [4.59, 4.RO] 0.527 40A 14 10 0.343 0.0750 -0.414 [-1.146, 0 .069] 4.36 [2. 74, 6.82] 0.354 41A 5 5 0.930 0.0081* -0.618 [-1.005, -0.331] 3.R4 [3.43, 4.15] 0.309 43A lR% L.l. 10 7 0.957 0.0001* -0.662 [ -0.839, -0.509]* 4. 79 [ 4.17. 5.50] 0.301 43A 25% L.J. 10 7 0.916 0.0007* -0.692 [-0.973, -0.470] 5.12 [ 4. 22, 6. 25] 0.296 43A 37% L. I. 10 7 0.940 0.0003* -0.668 [-0.886, -0.485} 5.09 [4.39, 5.94] 0.300 Table A.2. (continued) PCA Self-thinning Linee No. of PointsC Slone Interceot ld. Code a Conditionb "T r2 pd 8 95% CI f & 95% CI 0est q 43A 55% L. I. ll 8 0.934 <0.0001* -0.649 [-0.838, -0.487] 5. 22 4.59, 5. 95] 0.303 43A 100% L. I. 9 7 0.698 0.0192* -0.648 [-1.415, -0. 197] 5.30 3.55, 8. 27] 0.303 44A Ex. 2-60% L. I. 27 4 0.838 0.0844 -0.868 4.69 0.268 44A Ex. 2-100% L.I. 27 4 0. 7RO 0.1168 -0.926 5.14 0.260 44T Ex. l-23% L.I. 29 10 0.157 0.2576 -0.106 [-0.318, 0.0981 2.57 2. 14, 3.021 0.452 44T Ex. l-60% L.l. 32 R 0.834 0.0015* -0.313 [ -0.460, -0.176]* 3.60 3. 28, 3.%] 0.381 44T Ex. l-100% L. I. 32 6 0.364 0.2048 -0.147 [-0.453, 0. 133] 3.42 2 .69, 4.21] 0.436 44T Ex. 2-60% L.I. 27 4 0.842 0.0826 -0.867 4.71 0.268 44T Ex. 2-100% L.I. 27 4 0.764 0.1259 -0.968 5. 25 0.254 45A 32 25 0.391 0.0008* -0.475 [-0.775, -0.232] 3. 77 3.66, 3.R6] 0.339 4ilA 7 7 0.589 0.0439* -0.721 [-2.545, -0.052] 3.7? 3.53, 4.25] 0.291 49T 12 10 0.089 0.4031 -0.075 [-0.283, 0. 126] 3.03 2. 37. 3. 72] 0.465 oOT 15% L.I. 12 12 0.391 0.02% 0.284 0.037, 0.568] 0.97 0.19, 1.65 J 0.698 SOT 100% L.I. 22 22 0.021 0.5217 0.038 [ -0.085, 0. 162] 2. 27 l. 91' 2. 62] 0. 520 51 A 30 24 0.131 0.0819 -0.304 [-0.741, 0.048] 3.64 2.45, 5. 13] 0. 383 52 A 9 3 0.829 0.2711 -l. 594 6.52 0.193 53T 36 29 0.547 <0.0001* -0.391 [-0.541, -0.255] 4.31 [ 3.87' 4.80] 0.350 54 A 4 4 0.978 0.0113* -1.454 [ -2.504, -0.924]* 3. 35 [ 2. 63, 3.71] 0.204 N 55 A 17 11 0.333 0.0633 -1 .879 [ -0.596, 18.075] 9.69 [-66.11, 4.81] 0.174 0 56A 17 11 0.766 0.0004* -0.780 [-1.186, -0.488] 5.76 [ 4.65, 7. 32] 0.281 .;:::. 57 A 21 l3 0.559 0 .0033* -1.017 [-2.011, -0.51 9]* 6.58 [ 4.70, 10.321 0.248 SST 32 23 0.030 0.4304 -0.088 [-0.334, 0. 147] 2.50 [ 1.88, 3. 14 J 0.460 BOA 18 5 0.876 0.0192* -0.725 [ -1.506, -0.277] 4. 23 [ 3. 70, 4.541 0.290 SlA 7 4 0.886 0.0588 -0.615 [ -2.769, 0. 123] 4.06 [ 3.87, 4. 64 J 0.310 82A 42 17 0.468 0.0025* -0.305 [ -0.499, -0. 130]* 4.21 [ 4. ll' 4.30] 0.383 84A 286 243 0.248 <0. 0001 * -0.529 [ -0.653, -0.417] 3.73 [ 3.61, 3.84] 0.327 36A 18 15 0.582 0.0009* -0.985 f-l. 721' -0. 562]* 7.54 [ 5. 67. 10.80] 0.252 87A 15 12 0.934 <0.0001* -0.651 -0.781, -0.535]* 5.49 [ 5.05, 5.98] 0.303 88A 18 15 0.864 <0.0001* -0.993 [-1.265, -0.779]* 6.52 [ 5. 75, 7. 50] 0.251 89A Grazed 5 3 0.995 0.0447* -0.612 [-1.592, -0.089 4.14 [ 2 .09, 7 .99] 0.310 89A Ungrazed 5 5 0.948 0.0051* -0.618 [-0.932, -0.372 4.28 [ 3.36, 5.45] 0.309 90A Grazed 5 2 1.000 -0.870 5.67 0.267 oOA Ungrazed 5 2 1.000 -0.479 4.36 0. 338 91A 17% L. I. 25 13 0.464 0.0104* -0.146 [-0.252, -0.042]* 2.81 2.43, 3.20] 0.436 91A 100% L. I. 25 10 0.908 <0.0001* -0.427 [-0.543, -0.319] 4.80 4.36, 5. 27] o. 350 9lT 17% L. I. 20 l3 0.293 0.0559 -0.097 [-0.199, 0.003] 2.66 2.30, 3.04] 0.456 9lT 100% L. I. 20 10 0.854 0.0001* -0.245 [-0.330, -0.163]* 4.20 3.87, 4.54] 0.402 92A 17% L.I. 5 5 0. 760 0.0539 -0.189 [-0.398, 0.006] 2.69 l. 96, 3.48] 0.421 92A 23% L.l. 5 5 0.776 0.0483* -0.544 [-1.509, -0.011] 4.33 2.28, 8.06] 0.324 92A 44% L. I. 5 5 0.786 0.0450* -0.503 [-1.273, -0.027] 4.28 2.41l, 7. 20] 0.333 92A 100% L. I. 7 5 0.712 0.0725 -0.474 [-1.589, 0. 124] 4.3R 2.08, 8.67] 0.339 93A 8 R 0. 955 <0.0001* -0.954 [-1.189, -0.764]* 3.44 3.25, 3.60] 0.256 ? Table A.2. (continued) PCA Self-thinning Linee No. of PointsC Slope Intercept c~~~a Conditionb nT / pd il 95% Cl f & 95% CI Pest q 95A 3 2 1.000 -0.757 4.04 0.285 96A 28 4 0.892 0.0554 -1.262 3.02 0.221 97A 4 2 1.000 -0.841 3. 79 0.272 98A s. I. 22.8 4 2 1.000 -3.005 1.55 0.125 98A s. I. 28.9 4 4 0.971 0.0146* -2.478 [ -5.012, -1 .559]* 1.39 [-1.63, 2.48] 0.144 98A s. I. 33.5 4 4 0.964 0.0183* -1.066 [ -2.132, -0.549]* 3.44 [ 2.31, 3. 99] 0.242 99A 10 7 0.664 0.0255* -0.986 [ -3.077, -0.309] 2.70 [-0.84, 3.84) 0.252 lOlA 8 8 0.040 0.6340 -29.542 f -4.882, 7.398] -27.20 [-1.16, 11.80] 0.016 102A 10 9 0.939 <0.0001* -0.670 -0.837, -0.526]* 3.42 [ 3.23, 3.59] 0.299 103A 3 3 0.901 0.2038 -0.958 3.77 0.255 103T 3 3 0.949 0.1450 -0.782 3.97 0.281 l04A 6 6 0.945 0.0011* -0.467 -0.636, -0.318] 3.72 3. 51' 3. 91] 0.341 105A 3 3 0.861 0.2431 -0.218 4.07 0.411 1 05T 3 3 0.847 0.2560 -0.204 4.14 0.415 106A 8 8 0.770 0 .0042* -0.534 -0.889, -0.261] 3.46 3.43, 3.49) 0.326 109A 11 4 0.013 0.8861 0.060 3.48 0.532 llOA 12 8 0.280 0.1777 0.187 [ -0.125, 0.539] 4.04 3.72, 4.40) 0.615 llOT 4 3 0.780 0.3108 0.548 4.28 1.106 N lllA 12 10 0.848 0 .0002* -3.808 [ -5.766, -2.810]* 1.28 [-0.18, 2.02) 0.104 0 111T 12 10 0.842 0.0002* -3.760 [ -5.766, -2. 755]* 1.40 [-0.09, 2.14] 0.105 U""l ll2A 3 3 0.998 0.0270* -1.301 [ -2.495, -0.746]* 4.38 [ 4.26, 4.64) 0.217 l13A 6 5 0.887 0.0167* -1.326 [-3.173, -0.660]* 3.02 [ 1.09, 3. 72) 0.215 113T 6 5 0.878 0.0189* -1 .335 [ -3.430, -0.639]* 3.10 [ 0.91' 3.83] 0.214 114A 9 9 0.890 0.0001* -0.299 [ -0.396, -0.206]* 4.07 [ 4.05, 4. 10] 0.385 ll4T 9 9 0.887 0.0001* -0.299 [ -0.397, -0.205]* 4.16 [ 4. 13, 4.19] 0.385 115A 4 4 0.893 0.0552 -0.587 [ -2.050, 0.055] 4.04 [ 3.64, 4.22] 0.315 116A 4 3 1.000 0.0074* -0.513 [ -0.591' -0.439] 3.94 [ 3.85, 4.02) 0.330 117A 7 3 0.953 0.1384 -0.089 [ -0.361, 0.171] 4.47 [ 4. 16, 4.77] 0.459 118A 5 3 0.681 0.3819 -1.421 2.83 0.207 118T 5 3 0.650 0.4029 -1.389 2.91 0.209 119A 5 4 0.971 0.0144* -2.786 [ -5.645, -1. 772]* 1.71 [-0.49, 2.49) 0.132 120A 7 7 0.238 0.2665 -0.109 [ -0.352, 0.122] 4.01 [ 3. 93, 4. 10] 0.451 120T 7 7 0.339 0.1699 -0.133 [ -0.364, 0.084] 4.09 [ 4.01' 4. 16] 0.441 121A 9 8 0.575 0.0292* -0.509 [ -1.149, -0.087] 3.58 [ 3.09, 3.90] 0.331 121T 4 3 0.624 0.4200 -0.509 3. 70 0.331 122A 5 5 0.772 0.0499* -0.956 [ -20.004, -0.005] 3.84 [ 0. 56, 4.01] 0.256 123A 4 4 0.975 0.0125* -1.790 [ -3.279, -1.133]* 3.72 [ 3.41' 4.42) 0.179 124A 7 4 0.052 0. 7711 -0.031 [ -0.547, 0.468] 4.14 [ 4.03, 4.26] 0.485 125A 3 3 0.949 0.1450 -0.380 3.70 0.362 126A 6 5 0.988 0.0005* -0.586 [ -0.709, -0.474] 3.67 [ 3. 62, 3. 71] 0.315 128A 4 4 0.466 0.3174 -0.192 4.30 0.419 l30A 3 3 0.785 0.3066 -0.876 3.31 0.267 Table A.2. (continued) PCA Self-thinning Linee No. of Pointsc Slope Intercept !d. Condit ionb r2 pd 95% Cl f Code a nT n 8 & 95% CI Pest q l31A 6 6 0.956 0.0007* -0.498 -0.657, -0.356] 3.70 3.59, 3.81] 0.334 131T 4 4 0.953 0.0236* -0.224 -0.384, -0.075]* 4.11 3. 93, 4.28] 0.408 132A 8 8 0.206 0.2580 -0.294 3.86 0.386 l33A 8 8 0.566 0.0312* -0.188 [ -0.362, -0.024]* 3. 73 3.63, 3.82] 0.421 135A 5 5 0.854 0.0248* -3.047 [ -11.936, -1.628]* 4.63 3.54, 11.44] 0.124 136A 3 3 0.714 0.3593 -0.284 4.00 0.389 136T 3 3 0.763 0.3240 -0.331 4.05 0.376 137A 17 12 0. 983 <0.0001 * -0.422 [-0.462, -0.383]* 3.90 3.88, 3.92] 0.352 137T 17 12 0.982 <0.0001* -0.433 [ -0.476, -0.392]* 3.97 3.95, 3.99] 0.349 l38A 3 3 0.798 0.2969 -0.550 3.42 0.323 138T 3 3 0.783 0.3082 -0.544 3.61 0.324 aTable A.l associates the numeric part of each !d. code with a particular species and study. The letter indicates the type of biomass measurements made: A = aboveground parts only, T = aboveground and belowground parts both included. Dsee Table A.l and the references given there for further information on condition. CnT is the total number of log B-log N points reported for each code and condition. n the the number of points remaining after removing points not relevant to the thinning line. This is the number of points used to fit the PCA relationship between log B and log N. dThe statistical significance of the correlation between log B and log N. * indicates significance at the 951 confidence level (P ~ 0.05). ?Thinning lines were fitted to log B-log N data using principal component analysis (PCA). Forumlas for this analysis and for computing 951 confidence limits are given in Jolicoeur and Heusner (1971). f* indicates thinning slopes that are statistically different at the 95% confidence level from the thinning rule prediction of 8 = -1/2. gTransformed values of the log B-log N thinning slope, S, calculated from Pest = 0.5 I (1 - B) ( Chaoter 6) . .. N 0 0'1 207 Table A.3. Ranges of Log B and Log N Used to Fit Self-thinning Lines to Experimental and Field Data. Log Nc Log Be Id. Codea Conditionb n Min. Max. Mean Min. Max. Mean ,. 1A 3 0.08 1.20 0.61 3.27 3.76 3.51 1T 3 0.08 1.20 0.61 3.34 3.83 3.58 SA 4 -1.31 0.41 -0.14 3.50 3.97 3.68 7A 5 2.71 3.67 3.07 2.51 2.86 2.74 8A Lot 2B 7 -0.70 -0.16 -0.46 3.89 4.29 4.12 8A Lot 2C 7 -0.87 -0.43 -0.69 3.83 4.33 4.10 lOA Full light 10 3.05 4.74 3.87 2.36 3.17 2.77 14A 4 -0.91 -0.61 -0.77 4.12 4.31 4.23 15T 31 1.38 3.16 2.78 2.33 3.41 2.64 16T 38 2.47 4.05 3.29 2.01 2.88 2.43 17T 17 1.61 3.25 2.49 2.17 3.18 2.60 18T 43 1.24 3.17 2.43 2.09 3.11 2.49 19A 26 -1.06 1.38 0.07 3.51 5.08 4. 11 20T 13 2.98 5.12 3.87 3.12 3.98 3.58 21T 13 3.03 3.96 3.55 1.65 2.55 2.01 22T 34 -0.10 1.65 0.59 3.28 4.33 3.68 23T 23 -0.81 1.12 0.08 3.49 4.06 3.87 24T 5 0.50 1.39 0.88 3.79 4.18 3.98 26A 27 -0.75 0.16 -0.24 3.66 4.62 4.07 27A 2 2.83 3.12 2.98 3.35 3.44 3.40 28A Low L. I. 7 3.21 3.44 3.31 1.70 1.90 1.81 28A Med. L. I. 8 3.08 3.22 3.16 2.69 2.83 2.76 28A High L. I. 7 2.96 3.22 3.09 2.74 2.92 2.84 29T 9 2.80 3.75 3.16 2.44 3.10 2.90 30A 9 3.37 3.90 3.66 2.49 2.78 2.66 31A 11 3.00 3.74 3.46 2.28 2.62 2.46 32T 20% N.C. 5 3.46 4.01 3.66 2.64 2.84 2.77 32T 100% N.C. 5 3.09 3.90 3.42 3.03 3.17 3.12 33T 10 2.54 2.87 2.68 2.56 2.96 2.80 34T 10 2. 77 3.27 2.98 2.64 3.58 3.07 35A 13 3.14 3.80 3.46 2.78 3.21 3.01 36A 10 0.00 0.88 0.49 3.41 3. 77 3.54 38A 30% L. I. 7 2.80 3.91 3.24 1.99 2.26 2.16 38A 70% L. I. 8 2.67 3.88 3.19 2.43 3.07 2. 72 38A 100% L. I. 8 2.50 3.82 3.13 2.53 3.07 2.78 39A Kit ami 5 -0.86 -0.06 -0.48 4.82 5.24 5.06 39A Shirogane 7 -0.84 0.08 -0.34 4.56 4.79 4.67 40A 10 3.00 3.79 3.37 2. 71 3.16 2.97 41A 5 -1.25 -0.87 -1.07 4.38 4.61 4.50 43A 18% L.I. 7 3.61 4.41 4.03 1.82 2.44 2.12 43A 25% L.I. 7 3.68 4.42 4.03 2.02 2.65 2.33 43A 37% L. I. 7 3.47 4.46 3.88 2.16 2.80 2.50 208 Table A.3. (continued) Log Nc Log sc Id. Codea Conditionb n Min. Max. Mean Min. Max. Mean 43A 55% L.I. 8 3.46 4.49 3.88 2.30 2.99 2. 70 43A 100% L. I. 7 3.68 4.27 3.88 2.62 3.04 2.78 44A Ex. 2-60% L.I. 4 2.05 2.45 2.17 2.59 2.95 2.80 44A Ex. 2-100% L.I. 4 2.09 2.45 2.23 2.91 3.29 3.08 44T Ex. 1-23% L.I. 10 1.76 2.92 2.11 2.07 2.45 2.35 44T Ex. 1-60% L.I. 8 1.88 3.19 2.35 2.62 3.07 2.87 44T Ex. 1-100% L.I. 6 2.35 3.06 2.59 2.96 3.16 3.04 44T Ex. 2-60% L. I. 4 2.05 2.45 2.17 2.61 2.97 2.82 44T Ex. 2-100% L.I. 4 2.09 2.45 2.23 2.93 3.32 3.10 45A 25 -0.78 0.06 -0.38 3.62 4.24 3.95 48A 7 -0.05 0.52 0.29 3.27 3.76 3.51 49T 10 2.95 3.62 3.29 2.70 2.85 2.79 SOT 15% L. I. 12 2.17 3.34 2.76 1.46 2.05 1.76 50T 100% L. I. 22 2.03 4.00 2.91 2.08 2.56 2.38 51 A 24 2.86 3.76 3.41 2.35 2.93 2.61 52A 3 1.83 1.93 1.88 3.45 3.58 3.53 53T 29 2.92 3.64 3.28 2.84 3.23 3.03 54A 4 -0.81 -0.58 -0.69 4.17 4.52 4.35 55 A 11 3.66 3.92 3.80 2.40 2.82 2.55 56 A 11 3.64 3.95 3.82 2.68 2.96 2.78 57 A 13 3.51 3.92 3.77 2.55 2.94 2.74 58T 23 2.14 3.41 2.60 1.98 2.63 2.27 BOA 5 -0.78 -0.51 -0.68 4.62 4.83 4. 7 3 81A 4 -0.32 0.88 0.27 3.58 4.26 3.90 82A 17 -0.85 -0.05 -0.53 4.23 4.51 4.37 84A 243 -1.57 -0.26 -0.98 3.74 4.78 4.25 86A 15 4.09 4.83 4.43 2.85 3.54 3.18 87A 12 3.25 4.20 3. 77 2.75 3.45 3.04 88A 15 3.21 3.94 3.60 2.57 3.26 2.94 89A Grazed 3 3. 77 4.10 3.93 1.63 1.83 1.74 89A Ungrazed 5 3.55 4.00 3.73 1.81 2.09 1.97 90A Grazed 2 3. 77 3.94 3.85 2.24 2.39 2.31 90A Ungrazed 2 3.68 3.94 3.81 2.48 2.60 2.54 91A 17% L.I. 13 2.91 4.53 3.66 2.13 2.50 2.27 91A 100% L. I. 10 3.23 4.85 4.06 2.74 3.45 3.07 91T 17% L. I. 13 2.91 4.53 3.66 2.15 2.50 2.31 91T 100% L. I. 10 3.23 4.85 4.06 3.04 3.46 3.20 92A 17% L. I. 5 3.18 4.56 3.76 1.86 2.10 1.99 92A 23% L. I. 5 3.52 4.58 3.86 1.92 2.57 2.23 92A 44% L. I. 5 3.27 4.55 3.78 1. 92 2.69 2.38 92A 100% L. I. 5 3.37 4.57 3.85 2.13 2.78 2.56 93A 8 -1.14 -0.57 -0.82 3.96 4.56 4.22 95A 2 -0.16 0.16 0.00 3.92 4.16 4.04 209 Table A.3. (continued) Los N.c Log BC Id. Codea Conditionb n Min. Max. Mean Min. Max. Mean 96A 4 -1.19 -1 .04 -1.11 4.32 4.49 4.42 97A 2 -1. 18 - 1 ? 04 - 1. 11 4.66 4.78 4. 72 98A S.I. 22.8 2 -1.02 -0.98 -1.00 4.50 4.61 4.55 98A S.I. 28.9 4 -1.31 -0.97 -1.19 3.82 4.65 4.33 98A S.I. 33.5 4 -1.25 -0.82 -1.07 4.32 4.74 4.58 99A 7 -1.80 -1.46 -1.69 4.22 4.54 4.36 lOlA 8 -1.12 -0.91 -1.06 3.35 4.60 3.99 102A 9 -1.52 -0.85 -1.17 3.98 4.40 4.21 103A 3 -0.92 -0.59 -0.74 4.30 4.63 4.48 103T 3 -0.92 -0.59 -0.74 4.41 4.67 4.54 104A 6 -1.81 -0.84 -1.28 4.11 4.51 4.32 105A 3 -1.61 -0.44 -0.99 4.20 4.44 4.28 lOST 3 -1.61 -0.44 -0.99 4.26 4.49 4.34 106A 8 -0.17 0.52 0.10 3.14 3.62 3.40 109A 4 -0.31 0.23 -0.11 3.35 3.60 3.47 llOA 8 -1.46 -0.42 -1.01 3.64 4.02 3.85 llOT 3 -0.76 -0.42 -0.57 3.85 4.04 3.97 111A 10 -0.95 -0.50 -0.74 3.26 4.75 4.10 lllT 10 -0.95 -0.50 -0.74 3.35 4.82 4.19 l12A 3 0.08 0.35 0.22 3.92 4.27 4.10 113A 5 -1.26 -0.71 -1.05 4.07 4.80 4.41 113T 5 -1.26 -0.71 -1.05 4.16 4.88 4.50 114A 9 -0.92 3.00 0.29 3.26 4.44 3.98 l14T 9 -0.92 3.00 0.29 3.35 4.52 4.07 115A 4 -0.37 -0.05 -0.28 4.08 4.28 4.21 116A 3 -1.25 -1.06 -1.16 4.48 4.58 4.54 ll7A 3 -1.45 -0.91 -1.14 4.55 4.60 4.58 118A 3 -1.08 -0.78 -0.91 3.89 4.27 4.12 ll8T 3 -1.08 -0.78 -0.91 3.94 4.31 4.17 119A 4 -0.93 -0.56 -0.77 3.29 4.34 3.85 120A 7 -0.72 0.61 -0.35 3.91 4.17 4.05 120T 7 -0.72 0.61 -0.35 4.01 4.26 4.14 121A 8 -1.13 -0.30 -0.77 3.81 4.24 3.97 12lT 3 -1.06 -0.30 -0.74 3.91 4.34 4.08 122A 5 -0.44 0.00 -0.17 3.77 4.18 4.01 123A 4 o. 15 0.88 0.47 2.20 3.55 2.88 124A 4 -0.38 0.98 0.23 4.08 4.24 4.14 125A 3 -1.09 -0.28 -0.68 3.79 4.09 3.96 126A 5 -0.91 -0.20 -0.40 3.78 4.20 3.90 128A 4 0.90 1.94 1.50 3.88 4.17 4.02 l30A 3 -0.95 -0.28 -0.56 3.46 4.07 3.80 131A 6 -1.44 0.35 -0.72 3.49 4.32 4.06 131T 4 -1 ? 44 -0.7 3 -1. 12 4.26 4.42 4.36 210 Table A.3. (continued) Log NC Log BC Id. Code a Conditionb n Min. Max. Mean Min. Max. 132A 8 -0.92 0.00 -0.35 3.76 4.18 133A 8 -1.02 0.24 -0.56 3.62 3.93 l35A 5 0.60 0.99 0.77 1.60 2.90 136A 3 -1.52 -0.83 -1.19 4.21 4.42 l36T 3 -1.52 -0.83 -1.19 4.29 4.52 l37A 12 -1.07 0.14 -0.50 3.85 4.34 l37T 12 -1.07 0.14 -0.50 3.93 4.42 l38A 3 -0.34 0.70 0.07 3.03 3.65 138T 3 -0.34 0.70 0.07 3.22 3.84 arable A.l associates the numeric part of each Id. code with a particular species and study. The letter indicates the type of biomass measurements made: A = aboveground parts only, T = aboveground and be1owground parts both included. bsee Table A.l and the references given there for more information on condition. CThe mean, minimum, and maximum are given for log B and log N over the n data points used to fit the thinning line for each code and condition. .. Mean 3.96 3.83 2.29 4.34 4.44 4.11 4.19 3.38 3.57 211 Table A.4. Fitted Allometric Relationships for Experimental and Field Data. Allometric Relationship Fit by PCAc Id. 1\ 1\ Code a Conditionb n r2 pd 4>1 4>o height-weight allometry: log h = ~1 log w + ~0 (~1 = ~hw)e lA 3 0.997 0.0367* 0.282 -0.091 lT 3 0.996 0.0390* 0.283 -0.113 8A Lot 28 7 0.997 <0.0001* 0.384 -0.610 8A Lot 2C 7 0.998 <0.0001* 0.399 -0. 725 44A Ex. 2-60% L.I. 4 0.977 0 .0115* 0.368 -0.202 44A Ex. 2-1 00% L. I. 4 0.921 0.0404* 0.471 -0.394 44T Ex. 2-60% L.I. 4 0.975 0.0126* 0.368 -0.209 44T EX ? 2-1 00% L. I. 4 0.919 0.0416* 0.463 -0.397 48A 7 0.735 0.0137* 0.253 -0.086 81A 4 0.998 0.0011* 0.348 -0.587 86A 15 0.831 <0.0001* 0.303 0.016 87A 12 0.973 <0.0001* 0.356 -0.167 88A 15 0.899 <0.0001* 0.296 -0.212 93A 8 0.951 <0.0001* 0.339 -0.418 98A s. I. 28.9 3 1.000 0.0064* 0.389 -0.747 98A s. I. 33.5 3 0.993 0.0540 0.366 -0.576 99A 7 0.994 <0.0001* 0.337 -0.441 102A 9 0.928 <0.0001* 0.344 -0.544 103A 3 0.980 0.0892 0.328 -0.453 103T 3 0.993 0.0533 0.359 -0.640 104A 6 0.991 <0.0001* 0.195 0.191 105A 3 0.959 0.1292 o. 167 0.424 105T 3 0.959 0.1292 0.169 0.404 106A 8 0.977 <0.0001* 0.391 -0.712 110A 8 0.953 <0.0001* 0.334 -0.429 llOT 3 0.320 0.6170 0.251 -0.062 111A 10 0.931 <0.0001* 0.353 -0.619 lllT 10 0.935 <0.0001* 0.358 -0.676 112A 3 0.967 0.1154 0.279 -0.295 113A 5 0.938 0.0067* 0.239 -0.125 113T 5 0.939 0.0065* 0.238 -0.144 114A 9 0.974 <0.0001* 0.321 -0.512 ll4T 9 0.974 <0.0001* 0.321 -0.541 115A 4 0.983 0.0085* 0.163 0.253 116A 3 0.891 0.2137 o. 188 0.366 119A 4 0.997 0.0013* 0.446 -1.131 120A 7 0.954 0.0002* 0.297 -0.361 120T 7 0.954 0.0002* 0.291 -0.361 121A 8 0.970 <0.0001* 0.263 -0. 118 121T 3 1.000 0.0022* 0.243 -0.066 122A 5 0.958 0.0038* 0.387 -0.725 212 Table A.4. (continued) Allometric Relationship Fit by PCAc ld. 1\ 1\ Code a Conditionb n r2 pd l o height-weight allometry: log h = $1 log w + $0 ($1 = $hw}e 123A 4 0.974 0.0131 * 0.398 -0.471 124A 4 0.984 0.0082* 0.371 -0.718 125A 3 0.966 0.1189 0.436 -1.013 126A 5 o. 774 0.0492* 0.293 -0.554 128A 4 0.992 0.0038* 0.382 -1.130 130A 3 0.979 0.0930 0.365 -0.682 131A 6 0.978 0.0002* 0.288 -0.154 131T 4 0.982 0.0093* o. 184 0.390 132A 8 0.938 <0.0001* 0.322 -0.473 133A 8 0.962 <0.0001* 0.203 0.337 136A 3 0.991 0.0621 0.282 -0.179 136T 3 0.991 0.0613 0.272 -0.154 137A 12 0.985 <0.0001* 0.304 -0.278 137T 12 0.985 <0.0001* 0.302 -0.291 138A 3 1.000 0.0043* 0.228 -0.054 138T 3 1.000 0.0006* 0.229 -0.100 DBH-weight allometry: - 1\ 1\ IJ. A log D8H = <1>1 log w + o (11 = owle 8A Lot 28 7 0.999 <0.0001* 0.321 -2.334 8A Lot 2C 7 0.997 <0.0001* 0.328 -2.348 14A 4 0.995 0.0025* 0.274 -2.113 54A 2 1.000 0.278 -2.034 81A 4 1.000 0.0002* 0.395 -2.662 93A 8 0.994 <0.0001* 0.347 -2.482 95A 2 1.000 0.169 -1.754 96A 4 0.993 0.0037* 0.310 -2.099 97A 2 1.000 0.317 -2.364 98A s.r. 22.8 2 1.000 0.422 -2.946 98A s.r. 28.9 4 1.000 0.0002* 0.374 -2.687 98A s. I. 33.5 4 0.996 0.0020* 0.347 -2.556 99A 7 0.993 <0.0001* 0.392 -2.816 8SLA-weight allometry: - 1\ 1\ 1\ 1\ log 8SLA = <1>1 log w + o (l = 8wle 8A Lot 28 7 0.998 <0.0001* 0.651 -4.817 8A Lot 2C 7 0.999 <0.0001* 0.648 -4.759 14A 4 0.999 0.0006* 0.542 -4.303 45A 25 0.963 <0.0001* 0.826 -5.701 52 A 3 0.997 0.0344* o. 774 -5.259 54 A 4 0.997 0.0014* 0.664 -5.030 93A 8 0.992 <0.0001* 0.698 -5.089 213 Table A.4. (continued) Allometric Relationship Fit b~ PCAc Id. A $0 Code a Conditionb n r2 pd \Pl BSLA-weight allometry: - A log BSLA = 1P1 _ A A $ log w + ~Po (IPl = Bw}e 95A 2 1.000 0.765 -5.46 7 96A 4 0.997 0.0016* 0.719 -5.174 97A 2 1.000 0.633 -4.820 98A s. I. 22.8 2 1.000 0.839 -5.971 98A S.I. 28.9 4 1.000 0.0002* 0.746 -5.466 98A S.I. 33.5 4 0.996 0.0018* 0.692 -5.204 99A 7 0.999 <0.0001* 0.766 -5.630 1 OlA 8 0.949 <0.0001* 0.650 -5.040 102A 9 0.994 <0.0001* 0.697 -5.091 103A 3 0.978 0.0950 0.666 -5.106 103T 3 0.991 0.0590 0.728 -5.478 104A 6 1.000 <0.0001* 0.844 -6.071 105A 3 1.000 0.0021* 0.806 -5.813 lOST 3 1.000 0.0021* 0.815 -5.907 109A 4 0.994 0.0031* 1.231 -7.368 llOA 8 1.000 <0.0001* . 0.850 -5.856 llOT 3 0.992 0.0584 0.819 -5.763 lllA 10 0.988 <0.0001* 0.720 -5.314 lllT 10 0.989 <0.0001* 0.730 -5.425 112A 3 0.997 0.0328* 1 .045 -6.865 113A 5 0.992 0.0003* 0.887 -6.143 113T 5 0.991 0.0003* 0.886 -6.212 114A 7 0.997 <0.0001* 0.864 -5.891 114T 7 0.998 <0.0001* 0.858 -5.942 115A 4 0.995 0.0026* 0.845 -6.045 116A 3 1 .000 0.0067* 0.849 -5.883 117A 3 0.999 0.0204* 0.891 -6.190 118A 3 0.954 0.1379 1.030 -6.746 118T 3 0.960 0.1283 1 .051 -6.904 119A 4 0.997 0.0017* 0.951 -6.357 121A 8 0.961 <0.0001* 0.866 -6.033 121T 3 0.955 0.1358 0.878 -6.201 122A 5 1.000 <0.0001* 0.841 -5.836 123A 3 0.992 0.0564 1 .223 -6.684 124A 4 0.966 0.0172* 1 .567 -8.768 126A 5 0.995 0.0002* 0.809 -5.670 131A 6 0.994 <0.0001* 0.827 -5.789 131T 4 0.989 0.0053* 0.734 -5.360 132A 8 0.993 <0.0001* 0.871 -5.940 133A 8 0.989 <0.0001* 0.874 -5.961 l36A 3 1.000 0.0012* 0.822 -5.762 136T 3 1.000 0.0004* 0.794 -5.688 214 Table A.4. (continued) Id. Code a Conditionb n Allometric Relationship Fit bl PCAc r2 pd $1 A 1 0. 99* -0.62 Hutchings and Budd I OS] b 29T w Gra 25 Yoda et al. 1963 w Reg -0.48 4.41 White 1980 JOA w Reg 0. 76 -0.33 3. fl6 White ,and HarDer? I 970; White 1980 31A Reg 11 0.89 -0.42 3.93 White dnd Haroer l 970; White l 9RO 32T 20% N.C. w Ax 7 4.62 0.04 Furnas 1981 32T 100% N.C. w 1\x 7 4. 84 0.05 Furnas 1981 33T Restricted w Ax 5 4.07 0.04 Furnas l 981 Cont ro 1 w Ax 5 4 14 0.04 Furnas 1981 35A Reg 22 13 -0.56 [-0.79,-0.33] 4. 95 Westoby l g76 38A 70% L. I. Gr?a 20 Kays and Haroer 1974 38A 100% L.I. Gra 20 and Haroer 197~ 40A w 4 -0.46 1975 41 A w Gra 5 Bark ham 1978 t~3A lR% L. I. B Reg 10 7 0. 95 -0.67 0.18 and Howe 11 l 98 l 43A 25% L. I. B Reg 10 7 0.92 -0.66 0.07 and Howe 11 1981 43A 37% L.I. B Reg 10 I 0. 94 -0.65 0.08 and Howe 1 i 1981 43A 55% L. I. B Reg II 8 0. 93 -0.64 0.09 and Howe 11 I 981 43A 100% L.I. B Reg 9 7 0. 67 -0.56 0.07 and Howell I 981 44T Ex. l-60% l..l. w Reg 8 0. 96 -0.30 3. 53 White and Hdrper l 9/(] 14T Ex. l-1 00% I.. I. w Reg 9 0. 97 -0.33 3. 84 ~.~hite and Harper 1970 441 Ex. 2-60% L.l. w 6 0. 98 -0.70 4. 34 White and >0. 99* -0.79 [--1.19,-0.65] 4. 51 Lonsdale and I 98Ja 44T Ex. 2- l 00% L. I. Reg 0. 97 ??0. 84 4. 96 White and Hal~oer I 970 w f)CA >0. 99* -0.89 [ -1.29,-0.45] 5.06 Lonsdale and Watkin son 1 oR3a 45A w Reg -0.48 3. 53 Wnite l 980 217 Table A.5. (cant i nued) No. of Thinning Line Equation Points" 5looe Intercept !d. Oep. Fittin~ r2 or Code? Conditionb Var. c Method nT %Evf ?g 95% CI or SEh a SEh Reference i 48A w Reg -0.30 3.61 White 1980 51 A Low fert i 1 ity w Reg 15 -0.44 0.35 Bazzaz and Harper 1976 & High fertility w Reg 15 -0.64 0.19 Bazzaz and Harper 1976 52 A w Gra 9 Tadaki and Shidei 1959 53T w Reg 36 0.87 -0.41 4.19 White and Harper 1970; White 1980 54 A w Gra 4 Tadaki and Shidei 1959 55A w Reg 17 Malmberg and Smith 1982 56A w Reg 34 Ma 1 mberg and Smith 1982 57 A w Reg 21 0.87 -0.75 5.58 Malmberg and Smith 1982 82A w Gra Peet and Christensen 1980; Christensen and Peet 1982 84A w Gra Yoda et al. 1963 86A w PCA 18 15 >0. 99* -0.78 [-1.11,-0.52] 6.67 Lonsdale and Watkinson 1983a 87A w PCA 15 12 >0. 99* -0.68 [ -0.82,-0.56] 5.64 Lonsdale and Watkinson l983a 88A w PCA 18 15 >0. 99* -0.79 [ -1.08,-0.55] 5.82 Lonsdale and Watkinson 1983a 89A Grazed w Reg 5 3 -0.60 Oirzo and Harper 1980 89A Ungrazed w Reg 5 5 0.96 -0.80 Oi rzo and Harper 1980 90A Grazed w Gra 5 Oirzo and Harper 1980 90A Ungrazed w Gra 5 Dirzo and Harper 1980 91A 100% L. I. w PCA 25 >0.99* -0.43 [ -0.51,-0.35] 4.84 Lonsdale and Watkinson 1982 91T 100% L.J. w PCA 25 >0.99* -0.26 [ -0.33,-0.19] 4.29 Lonsdale and Watkinson 1982 92A 23% L. I. w PCA 5 >0.99* -0.52 [ -0.74,-0.30] 4.26 Lonsdale and Watkinson 1982 92A 44% L.I. w PCA 5 >0.99* -0.49 [ -0. 71,-0.27] 4.26 Lonsdale and Watkinson 1982 92A 100% L.I. w PCA 5 >0.99* -0.46 [ -0.73,-0.19] 4. 37 Lonsdale and Watkinson 1982 'Table A.l associates the numeric part of each Id. code with a Particular species and study. The letter indicates the type of biomass measurements made: A = aboveground parts only, T = aboveground and belowground parts both included. "+" indicates that the 1 i ne is a dup 1 i cate analysis of the previous code and condition by different author. 11 &11 indicates a treatment or condition analyzed separately in the reference above, but combined with the preceding condition in the reanalysis of this report (Table A.2). bsee Table A. 1 and the references given there for further information on condition. C"W" indicates that the data were analyzed with log w-log N plots or by fitting log w against log N, while "B" indicates that log B-log N plots or fitting was used. dMethod of estimating the thinning line. "Gra" indicates graphical analysis only with no statistics, "Reg" indicates regression analysis, and 11 PCA 11 indicates principal component analysis. 11 Ax'' indicates a method used by Furnas (1981), who assumed a priori that thinning slopes were -1.5 in the log w-log N plane, then used statistics to estimate the intercept under this-axiom. Blanks indicate that the method of fitting was not specified. enT is the total number of size-density points reported for each code and condition. n the the number of points remaining after removing points not relevant to the thinning line. This is the number of points used to fit the relationship between size and density. Blanks indicate that sample sizes were not given in the reference. fThis column contains coefficients of determination if the reference analyzed the data by correlation or regression. If the analysis was done by PCA, the number here is the percentage of the combined variance of the two variables explained by the first principal component (%EV). gMost thinning slopes were originally given for data in log w-log N form. Such slopes were converted here to log B-log N form by adding 1 (Chapter 1). This facilitates comparison with the analyses in Table A.2. hsingle numbers in the column are standard errors. Paired numbers in brackets are 95% confidence limits. iThese are references for the self-thinning analysis only. References to the actual data are in Table A.l *Percent variance explained (see footnote f). 218 APPENDIX B FORESTRY YIELD TABLE DATA Table B.l. Sources of Forestry Yield Table Data. Wood !d. Shade Dens1tY. Referencef Code? Species Groupb ToleranceC (kg/m )a Conditions? 201 Pi nus oonderosa G 472 S. I.{ft) 100 Meyer 1938, Tabs. 3-6 202 Alnus~ A 480 S.I.(ft) 50 Smith 1968, Tab. 3 203 Ab 1 es conco lor G 360 s. I. (ft) 50 Schumacher 1926, Tab. 204 Ca:stanea dentata A 737 S.I.{ft) 50 Frothingham 1912, Tabs. 17-19 205 Castanea dentata and A 737 S.I.(ft) 50 Frothingham 1912, Tabs. 20-22 Quercus-g--- 206 Quercus9 A 3 737 S. I. {ft) 50 Frothingham 1912, Tabs. 23-25 207 ca;:::yag- A 2 657 S.I.{ft) 50 Boisen and Newl.i n 1910, Tab. 14 208 P1nus monticola G 3 432 S.!.(ft) 50 Haig 1932, Tab. l 209 Populus tremuloides A l 400 S.Q. {Ranks) Baker 1925, Tabs. 14-17 210 Populus deltoides A l 384 s. I. (ft) 50 Williamson 1913, Tab. 3 211 ChamaeC,lPans thyoides G 4 368 S.I.(ft) 50 Korstian 1931, Tabs. 22,25 212 Picea sitchens1s and G 4 472 s. I. (ft) 100 Meyer 1937, Tabs. l-5 Tsuga heterophyll a 213 Picea rubrens G 448 S.I.{ft) 50 Meyer 1929, Tabs. 2-6 214 Pinusg--- G 513 S.Q. (Ranks) Khil 'mi 1957, Tabs. 13,14 215 P1cea9 G 416 S.Q. (Ranks) Khil 'mi 1957, Tabs. 18,45 216 Quercus9 A 737 S.Q. {Ranks) Khil 'mi 1957, Tabs. 23,50 217 Pinus strobus G 3 400 S.Q. (Ranks) Marty 1965, Tab. l 218 Pi nus strobus G 3 400 S.Q. {Ranks) Marty 1965, Tab. 2 220 Pinus strobus G 3 400 S.Q. (Ranks) Marty 1965, Tab. 4 221 Pseudotsu~a menzies i i G 3 513 S.I.{ft) 100 McArdle 1930, Tab. 12 222 Pinus res1nosa G 2 529 Spacing (Ranks) Stiell and Berry 1973, Tabs. 4-8 223 Pinusg--- G 513 S.Q. (Ranks) Tseplyaev 1961, Tab. 47 ' N 224 Cedrus9 G 448 S.Q. (Ranks) Tseplyaev 1961, Tab. 81 ...... 225 Populus9 A 400 S.Q. {Ranks) Tseolyaev 1961, Tab. 134 1..0 227 Pinus Ponderosa G 2 472 s. I. (ft) 100 Behre 1928, Tabs. 2-6 228 Thuja occ1dentalis G 4 304 S.I.(ft) 160 Gevorkiantz and Duerr 1939, Tabs. 18-24 229 P1nus taeda G 2 609 S.I.(ft) 50 Schumacher and Coile 1960, Tab. 230 Pinus eTTIOttii G 3 609 S.I.{ft) 50 Schumacher and Coile 1960, Tab. 231 Pinus palustris G l 609 S.I.(ft) 50 Schumacher and Coile 1960, Tab. 232 Pinus ech1nata G 2 609 S.I.(ft) 50 Schumacher and Coile 1960, Tab. 233 P1nus serot1na G 2 609 S.I.(ft) 50 Schumacher and Coile 1960, Tab. 234 Pinus~ G 2 609 S.I.(ft) 50 Ashe 1915, Tabs. 19,35,42 235 P1nus palustris G l 609 S.l.(ft) 50 Wahlenberg 1946, Tabs. 4b,6c,7a,7,b 236 Quercus9 (Upland oaks) A 3 737 S.I.{ft) 50 Schnurr 1937, Tab. 2 237 Picea gl;uca G 4 400 Soacing (Ranks) Stiell 1976, Tabs. 10-13 238 Ab1es ba samea G 5 416 S.I.(ft) 65 Meyer 1929, Tabs. 35-49 239 P1cea glauca G 4 400 S.I.{ft) 65 Meyer 1929, Tabs. 29-33 241 Picea9 and Abiesg G 4 416 S.I.(ft) 50 Bakuzis and Hansen 1965, Tabs. 89,91,93,95,97 242 Alnus rubra A 480 S.I.{ft) 50 Smith 1968, Tab. 5 243 Alnus rubra A 480 Crown width Smith 1968, Tab. 6 244 Southe;::;;RiTxed hardwoods A 641 S.I.(ft) 50 Frothingham 1931, Tab. 8 245 Northern mixed hardwoods A 641 S.Q. (Ranks) Gevork iantz and Duerr 1937, Tab. 3 246 Pooulus9 {Aspen} A 464 S.I.(ft) 50 Kittredge and Gevorkiantz 1929, Tab. 247 Eucalyptus globut E 889 S.Q. (Ranks) Jacobs 1979, Tab. A3.5 248 Eucalyptus m1cro heca E 801 S.Q. (Ranks) Jacobs 1979, Tab. A3.l3 249 L iriodendron tulipifera A 2 448 S.I.(ft) 50 McCarthy 1933, Tab. 17 250 Frax1nusY A 3 641 S.Q. (Ranks) ~~f~~m !i 1 ~l.Tf~49; 5rab. 251 ~uga menziesii G 3 5l3 none 25 Table B.l. (cant i nued) ld: Code a 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 276 277 278 279 280 281 Wood Shade Species Groupb To 1 erancec Dens1tY. (kg/m )0 Conditionse Referencef Sequoia semoervirens G 4 416 S.l. (ft) 100 Lindquist and Palley 1963, Tabs. 1 ,2 ,4-6 Pinus echinata G 2 609 S.l.(ft) 50 Mattoon 1915, Tab. 14 Pinus echinata G 2 609 S.l.(ft) 50 Mattoon 1915, Tab. 16 Picea sitchensis G 4 424 S.l. (ft) 50 Cary 1922, Tabs. 6-9 P1 nus banks i ana G 1 464 S.I. (ft) 50 Eyre 1944, Tab. 8 Pinus banksiana G 1 464 S.l. (ft) 50 Bella 1968, Tabs. 2-6 Pinus resinosa G 2 529 S.I.(ft) 50 Eyre and Zehnqraff 1948, Tab. 10 tsuga heteroph;tll a G 5 521 S.I.(ft) 100 Barnes 1962, Tabs. 3,5,7,9, 12 ~ heterophyl1 a G 5 521 S.I.(ft) 100 Barnes 1962, Tabs. 4,6,8, 10,13 Tsuga heterophyll a G 5 521 S.l. (ft) 100 Barnes 1962, Tabs. 4,6,8, 10,14 P1cea s1tchens1s and G 4 472 S.l.(ft) 100 Taylor 1934, Tabs. 4-A Tsuga heterophyll a Picea mariana G 529 S.l.(ft) 50 Fox and Kruse 1939, Tab. 2 Pwus banks1ana G 464 S.T. (ft) 50 Boudoux 1978, Tabs. 3,4 Fraxinus americana A 657 S.I.(ft) 50 Patton 1922, page 37 Pseudotsugamenz;es i i G 512 S.l.(ft) 50 Schumacher 1930, Tabs. 2-5' 7 Northern m1 xed hardwoods A 641 S.l.(ft) 50 Forbes 1961, Tab. 14 Northern mixed hardwoods A 641 S.l. (ft) 50 Vermont Ag. Expt. Sta. 1914, Tab. 8 Quercusg (Red oaks) A 737 S.l. (ft) 50 Patton 1g22, paqe 37 P1nus taeda G 609 S.l.(ft) 50 USDA 1929, Tabs. 33-38 Pinus palustris G 609 S.l. (ft) 50 USDA 1929, Tabs. 68-70 Pinus ponderosa G 472 S.l.(ft) 100 Show 1925, Tab. 2 Pinus echinata G 609 S.I.(ft) 50 USDA 1929, Tabs. 98-102 Pinus elhottii G 609 S.l.(ft) 50 USDA 1929, Tabs. 130-134 Sequoia sempervirens G 416 S.I.(ft) 50 Bruce 1923, Tabs. 1-3 L 1qu1damber styrac1flua A 593 S.l.(ft) 50 Winters and Osborne 1935, Tabs. 4-A Tsuga heteroph,~:lla G 520 S. I.(ft) 100 Barnes 1962, Tab. 27 Eucalyptus de1egatensis E 617 S.l. (m) 50 Hillis and Brown 1978, Tab. 10.1 Eucalyptus regnans E 689 S.l.(m) 50 Hillis and Brown 1978, Tab. 10.22 Eucalyptus s ieberi E 840 S.I.(m) 50 Hillis and Brown 1978, Tab. 10.27 ?An arbitrary number assigned to f ac i 1 it ate cross-referencing among tab 1 es. 0Qne of three categories. A = temperate angiosperms, E = trees of genus Eucalyptus, Cfrom references above or Baker ( 1949). temperate qymnosperms. dFrom references above or Peatt i e ( 1950, 1953). Used to convert stand vo 1 umes (m3 of wood per m2 of ground area) to stand biomasses (grams of wood per m2). ?Differences in growing site or cultural practice that may affect thinning line. S.l. =height of trees (in either feet or meters, as indicated) when the stand is at the specified age (in years). This measure, called 'site index', is a widely used indicator of the quality of a site for tree growth. S.Q. =Site quality measured by ranks rather than by site index. Lower numbers are of lower quality. Spacing= ranked initial densities in tree plantations, with lower numbers indicating higher initial densities. Crown width = a ranked measure of canopy openness. Lower numbers indicate a more open canopy. See references for further details. foata are taken from the indicated tables in the reference. gGeneric name only indicates that the specific name was not given or t: ?? t more than one species from the genus were present. N N 0 221 Table B.2. Self-thinning Lines Fit to Forestry Yield Table Data. No. of Pointsc I d. PCA Thinning Linee Log Nf Log sf Codea Cond.b nT n rd s & Pest Min. Max. Mean Min. Max. Mean 201 40 18 13 -0.998 -0.308 3.75 0.382 -1.27 -0.49 -0.97 3.90 4.14 4.05 201 50 19 14 -0.997 -0.310 3.77 0.382 -1.38 -0.52 -1.07 3.93 4.21 4.11 201 60 19 14 -1.000 -0.324 3.79 0.378 -1.49 -0.69 -1.18 4.01 4.27 4.17 201 70 19 14 -0.999 -0.326 3.83 0.377 -1.57 -0.81 -1.27 4.09 4.34 4.25 201 80 19 14 -0.999 -0.321 3.89 0.378 -1 .64 -0.91 -1.35 4.17 4.41 4.32 201 90 19 14 -0.998 -0.318 3.95 0.379 -1.71 -0.99 -1.42 4.25 4.49 4.40 201 100 19 14 -0.997 -0.314 4.01 0.380 -1 .76 -1 .06 -1.47 4.33 4.56 4.47 201 110 19 14 -0.997 -0.307 4.08 0.383 -1.80 -1.12 -1.52 4.41 4.63 4.55 201 120 19 14 -0.993 -0.318 4.12 0.379 -1.84 -1.17 -1 .56 4.48 4.70 4.62 201 130 19 14 -0.988 -0.318 4.18 0.379 -1.87 -1.22 -1.59 4.55 4.77 4.69 201 140 9 4 -0.999 -0.498 3.98 0.334 -1.46 -1.26 -1.37 4.61 4.71 4.66 201 150 9 4 -0.998 -0.479 4.04 0.338 -1.49 -1.30 -1.40 4.66 4.75 4.71 201 160 9 4 -0.998 -0.524 4.01 0.328 -1 .52 -1.33 -1.43 4.71 4.81 4.76 202 80 11 9 -0.999 -0.653 3.30 0.303 -1.38 -0.76 -1.15 3.79 4.19 4.05 202 96 11 9 -0.998 -0.470 3.59 0.340 -1.46 -0.86 -1.24 3.98 4.27 4.17 202 112 11 9 -0.999 -0.484 3.62 0.337 -1.54 -0.94 -1.31 4.07 4.37 4.25 203 30 11 8 -1.000 -0.563 3.67 0.320 -0.83 -0.59 -0.73 4.00 4.14 4.08 203 40 11 8 -0.998 -0.575 3.69 0.317 -0.96 -0.73 -0.87 4.10 4.24 4.19 203 50 11 8 -1 .000 -0.585 3.77 0.315 -1 .05 -0.83 -0.96 4.25 4.39 4.33 203 60 11 8 -1.000 -0.564 3.89 0.320 -1.14 -0.91 -1.05 4.40 fl.54 4.48 203 70 11 8 -0.999 -0.558 3.95 0.321 -1.23 -0.99 -1.13 4.51 4.64 4.59 203 80 11 8 -0.999 -0.570 3.98 0.318 -1.31 -1.07 -1.21 4.58 4. 72 4.67 203 90 11 8 -0.999 -0.580 3.96 0.316 -1.38 -1.15 -1.29 4.63 4.77 4.71 204 62 13 7 -0.997 -0.640 3.56 0.305 -1.14 -0.91 -1.04 4.13 4.29 4.22 204 71 13 8 -0.998 -0.638 3.64 0.305 -1.23 -0.92 -1.09 4.22 4.42 4.33 204 80 13 8 -0.998 -0.657 3.67 0.302 -1.31 -1.01 -1.17 4.32 4.52 4.43 205 58 13 8 -0.997 -0.766 3.44 0.283 -1.02 -0.76 -0.90 4.02 4.21 4.13 205 68 13 8 -0.997 -0.691 3.56 0.296 -1.15 -0.88 -1.03 4.16 4.35 4.27 205 80 13 8 -0.999 -0.630 3.67 0.307 -1.24 -0.95 -1.11 4.27 4.45 4.37 206 52 l3 9 -0.996 -0.807 3.31 0.277 -1.07 -0.72 -0.92 3.89 4.16 4.05 206 64 13 9 -0.996 -0.747 3.41 0.286 -1.20 -0.85 -1.04 4.03 4.29 4.19 206 75 l3 9 -0.998 -0.698 3.50 0.294 -1.28 -0.93 -1.14 4.15 4.39 4.29 207 49 11 11 -0.992 -0.787 2.94 0.280 -1.79 -0.76 -1.36 3.57 4.42 4.01 208 40 8 6 -0.997 -0.795 4.01 0.279 -0.66 -0.13 -0.49 4.11 4.55 4.39 208""" 50 8 6 -0.997 -0.803 3.94 0.277 -0.84 -0.31 -0.66 4.18 4.63 4.47 208 60 8 6 -0.998 -0.796 3.83 0.278 -1.07 -0.53 -0.89 4.25 4.69 4.54 208 70 8 6 -0.997 -0.766 3.75 0.283 -1.27 -0.73 -1.10 4.31 4.75 4.59 209 1 7 4 -0.976 -0.318 3.46 0.379 -1.46 -0.97 -1.20 3.76 3.92 3.85 209 2 9 6 -0.937 -0.282 3.63 0.390 -1.61 -0.89 -1.20 3.85 4.05 3.97 209 3 11 7 -0.947 -0.270 3.74 0.394 -1.57 -0.92 -1.21 3.96 4.14 4.07 209 4 13 8 -0.985 -0.433 3.66 0.349 -1.33 -0.95 -1.16 4.05 4.23 4.16 210 136 44 28 -0.998 -0.199 3.85 0.417 -2.10 -1.49 -1.83 4.15 4.27 4.22 211 20 17 14 -0.998 -0.537 3.53 0.325 -0.42 0.26 -0.17 3.38 3.74 3.62 211 30 17 14 -0.997 -0.529 3.76 0.327 -0.53 0.16 -0.27 3.67 4.02 3.90 222 Table B.2. (continued) No. of Log Nf Log Bf Pointsc PCA Thinning Linee Id. Code a Cond.b nT n rd s & Pest Min. Max. Mean Min. Max. Mean 211 40 17 14 -0.999 -0.510 3.86 0.331 -0.65 0.05 -0.40 3.83 4.19 4.07 211 50 17 14 -0.999 -0.535 3.89 0.326 -0.81 -0.14 -0.56 3.96 4.32 4.19 211 60 17 14 -0.998 -0.534 3.88 0.326 -1 .02 -0.34 -0.76 4.05 4.42 4.29 211 70 17 14 -0.998 -0.535 3.84 0.326 -1.25 -0.56 -0.99 4.13 4.49 4.37 212 60 18 14 -0.994 -0.509 3.95 0.331 -1 .06 -0.50 -0.83 4.18 4.48 4.37 212 80 19 15 -0.993 -0.551 3.97 0.322 -1.19 -0.55 -0.93 4.24 4.61 4.49 212 100 19 15 -0.993 -0.542 4.03 0.324 -1.34 -0.68 -1 .08 4.37 4.74 4.61 212 120 19 15 -0.994 -0.554 4.05 0.322 -1.50 -0.84 -1.24 4.49 4.86 4.73 212 140 19 15 -0.992 -0.572 4.03 0.318 -1.65 -1.03 -1.41 4.59 4.96 4.84 212 160 19 15 -0.991 -0.556 4.06 0.321 -1.80 -1.17 -1.55 4.67 5.04 4.92 212 180 19 15 -0.992 -0.586 3.99 0.315 -1.92 -1.31 -1.68 4.72 5.10 4.97 212 200 19 15 -0.993 -0.565 3.99 0.319 -2.06 -1.43 -1.81 4.77 5.14 5.01 213 30 10 6 -0.998 -0.545 3.72 0.324 -0.38 -0.11 -0.29 3.78 3.93 3.88 213 40 10 6 -0.996 -0.536 3.83 0.325 -0.65 -0.39 -0.57 4.04 4.18 4.13 213 50 10 6 -0.998 -0.542 3.88 0.324 -0.81 -0.54 -0.73 4.18 4.32 4.27 213 60 10 6 -0.997 -0.538 3.93 0.325 -0.91 -0.64 -0.82 4.28 4.42 4.37 213 70 10 6 -0.997 -0.540 3.97 0.325 -0.98 -0.71 -0.90 4.36 4.50 4.45 214 1 11 6 -0.998 -0.636 3.79 0.306 -0.96 -0.64 -0.82 4.19 4.39 4.31 214 2 13 9 -0.997 -0.638 3.79 0.305 -1.20 -0.73 -1 .01 4.25 4.55 4.44 214 3 13 9 -0.997 -0.622 3.83 0.308 -1.33 -0.87 -1.15 4.36 4.65 4.54 214 4 13 9 -0.998 -0.625 3.85 0.308 -1.40 -0.97 -1.23 4.45 4.73 4.62 214 5 13 9 -0.999 -0.606 3.91 0.311 -1.45 -1.03 -1.29 4.53 4.79 4.69 214 6 13 9 -0.999 -0.608 3.95 0.311 -1.50 -1 .09 -1.34 4.60 4.86 4.77 215 1 9 9 -0.999 -0.994 3.65 0.251 -0.75 o. 11 -0.46 3.53 4.37 4.10 215 2 10 10 -0.998 -0.964 3.66 0.255 -0.91 0.23 -0.51 3.40 4.50 4.15 215 3 11 11 -0.995 -0.971 3.65 0.254 -1.04 0.45 -0.55 3.12 4.61 4.18 215 4 11 11 -0.996 -0.939 3.68 0.258 -1.15 0.27 -0.68 3.36 4.72 4.32 215 5 11 11 -0.997 -0.905 3.74 0.262 -1.22 0.07 -0.79 3.62 4.81 4.45 215 6 11 11 -0.998 -0.904 3.76 0.263 -1.28 -0.08 -0.90 3.79 4.90 4.58 216 1 11 11 -1.000 -0.729 3.65 0.289 -1.48 0.04 -0.96 3.61 4.72 4.35 216 2 11 11 -0.999 -0.739 3.66 0.288 -1.59 -0.17 -1.10 3.76 4.81 4.47 216 3 11 11 -0.998 -0.747 3.66 0.286 -1.68 -0.32 -1.21 3.87 4.89 4.56 217 54 10 6 -0.997 -0.562 3.77 0.320 -1.27 -0.72 -1.03 4.16 4.47 4.34 217 64 10 6 -0.997 -0.544 3.82 0.324 -1.35 -0.88 -1.14 4.29 4.55 4.44 217 75 10 6 -0.996 -0.528 3.87 0.327 -1.42 -1.00 -1.23 4.39 4.61 4.53 218 50 10 10 -0.997 -0.776 3.58 0.281 -1.09 -0.16 -0.79 3.69 4.40 4.19 218 60 10 10 -0.999 -0.803 3.59 0.277 -1.19 -0.27 -0.90 3.80 4.53 4.32 218 70 10 10 -0.998 -0.788 3.65 0.280 -1.26 -0.35 -0.97 3.91 4.62 4.41 220 40 9 6 -0.995 -1.065 3.43 0.242 -0.60 -0.31 -0.47 3.74 4.07 3.93 220 50 9 6 -0.992 -0.985 3.50 0.252 -0.72 -0.40 -0.57 3.88 4.20 4.07 220 60 9 6 -0.991 -0.879 3.59 0.266 -0.85 -0.49 -0.68 4.00 4.32 4.19 220 70 9 6 -0.987 -0.746 3.69 0.286 -0.99 -0.56 -0.78 4.09 4.41 4.28 220 80 9 6 -0.984 -0.623 3.79 0.308 -1.15 -0.64 -0.90 4.16 4.48 4.35 220 90 9 6 -0.979 -0.528 3.88 0.327 -1.30 -0.71 -1.00 4.22 4.54 4.41 223 Table 8.2. (continued) No. of Pointsc PCA Thinning Linee Log Nf Log Bf Id. Codea Cond.b nT n rd il & Pest Min. Max. Mean Min. Max. Mean 221 80 15 13 -0.996 -0.629 3.64 0.307 -1.21 -0.42 -0.94 3.88 4.38 4.23 221 90 15 13 -0.995 -0.655 3.66 0.302 -1.25 -0.50 -1.00 3.96 4.46 4.31 221 100 15 13 -0.996 -0.667 3.68 0.300 -1.31 -0.57 -1.05 4.03 4.53 4.38 221 110 15 13 -0.995 -0.684 3.69 0.297 -1.36 -0.64 -1.11 4.10 4.60 4.45 221 120 15 13 -0.994 -0.683 3.72 0.297 -1.43 -0.71 -1.17 4.17 4.67 4.52 221 130 15 13 -0.993 -0.695 3.72 0.295 -1.48 -0.77 -1.23 4.23 4.73 4.58 221 140 15 13 -0.992 -0.693 3.73 0.295 -1.54 -0.84 -1.29 4.27 4.77 4.62 221 150 15 13 -0.992 -0.697 3.73 0.295 -1.60 -0.90 -1.35 4.31 4.81 4.66 221 160 15 13 -0.990 -0.696 3.72 0.295 -1.65 -0.96 -1.40 4.34 4.84 4.69 221 170 15 13 -0.990 -0.694 3. 71 0.295 -1.72 -1.02 -1.46 4.37 4.87 4.72 221 180 15 13 -0.992 -0.696 3.68 0.295 -1.78 -1.08 -1 .53 4.39 4.89 4.74 221 190 15 13 -0.991 -0.700 3.64 0.294 -1.85 -1.16 -1.60 4.41 4.91 4.76 221 200 15 13 -0.992 -0.705 3.60 0.293 -1.93 -1.23 -1.67 4.43 4.93 4.78 221 210 15 13 -0.991 -0.691 3.59 0.296 -2.01 -1.30 -1.75 4.45 4.95 4.80 222 1 35 8 -0.997 -0.631 4.21 0.307 -0.66 -0.44 -0.52 4.48 4.62 4.54 222 2 35 8 -0.997 -0.880 3.98 0.266 -0.72 -0.54 -0:61 4.45 4.61 4.51 222 3 35 8 -0.996 -1.242 3.62 0.223 -0.78 -0.64 -0.69 4.42 4.59 4.49 .. 222 4 35 5 -0.999 -1.535 3.27 o. 197 -0.85 -0.77 -0.80 4.44 4.57 4.50 222 5 35 5 -0.995 -1.975 2.73 0.168 -0.92 -0.86 -0.88 4.42 4.55 4.47 223 1 15 13 -0.999 -0.338 4.02 0.374 -1.59 -0.84 -1.35 4.30 4.55 4.48 224 3 17 14 -0.995 -0.504 3.91 0.333 -1.36 -0.61 -1.14 4.23 4.60 4.49 225 1 6 6 -0.998 -0.687 3.40 0.296 -1.32 -0.56 -1.02 3.78 4.31 4.09 227 40 16 13 -0.994 -0.511 2.99 0.331 -1.41 -0.79 -1.17 3.41 3.73 3.59 227 50 16 13 -0.995 -0.508 3.26 0.331 -1.46 -0.83 -1.22 3.70 4.02 3.88 227 60 16 13 -0.995 -0.508 3.41 0.332 -1 .51 -0.88 -1.27 3.88 4.20 4.06 227 70 16 13 -0.995 -0.509 3.50 0.331 -1.57 -0.94 -1.33 4.00 4.32 4.18 227 80 16 13 -0.994 -0.509 3.55 0.331 -1.63 -1 .01 -1.39 4.07 4.39 4.26 227 90 16 13 -0.994 -0.509 3.57 0.331 -1.70 -1.07 -1.46 4.13 4.45 4.31 227 100 16 13 -0.993 -0.508 3.60 0.332 -1.77 -1.14 -1 .53 4.20 4.52 4.38 227 110 16 13 -0.994 -0.509 3.64 0.331 -1.84 -1.21 -1.60 4.27 4.59 4.45 227 120 16 13 -0.977 -0.509 3.67 0.331 -1.91 -1.28 -1 .68 4.35 4.67 4.53 228 25 14 11 -0.998 -0.407 3.46 0.355 -0.55 0.06 -0.34 3.42 3.68 3.59 228 35 14 11 -0.998 -0.372 3.52 0.364 -0.75 -0.11 > -0.53 3.56 3.80 3.72 228 45 15 12 -0.997 -0.368 3.56 0.366 -0.93 -0.10 ?-0.65 3.58 3.89 3.80 228 55 15 12 -0.998 -0.340 3.60 0.373 -1 .07 -0.24 -0.79 3.67 3.96 3.87 228 65 15 12 -0.998 -0.319 3.64 0.379 -1.21 -0.37 -0.93 3.75 4.02 3.94 228 75 15 12 -0.998 -0.301 3.67 0.384 -1.34 -0.49 -1 .05 3.81 4.06 3.98 229 60 7 7 -0.999 -0.892 3.05 0.264 -1.23 -0.80 -1.09 3.76 4.14 4.02 229 70 7 7 -0.999 -0.898 3.05 0.263 -1.33 -0.91 -1.19 3.86 4.24 4.12 229 80 7 7 -0.997 -0.895 3.07 0.264 -1.41 -0.99 -1.27 3.94 4.32 4.21 229 90 7 7 -0.997 -0.887 3.11 0.265 -1.48 -1.05 -1.34 4.03 4.41 4.30 229 100 7 7 -0.995 -0.888 3.13 0.265 -1.55 -1.12 -1.40 4.11 4.49 4.38 229 110 7 7 -0.992 -0.884 3.17 0.265 -1.60 -1.17 -1.45 4.18 4.56 4.45 229 120 7 7 -0.988 -0.880 3.21 0.266 -1.64 -1.21 -1.49 4.26 4.63 4.53 224 Table B.2. (continued) No. of PCA Thinning Linee Pointsc Id. Log N f Log Bf Codea Cond.b nr n rd i3 & Pest Min. Max. Mean Min. Max. Mean 230 50 7 7 -0.999 -0.565 3.36 0.320 -1.09 -0.49 -0.90 3.64 3.98 3.87 230 60 7 7 -0.998 -0.602 3.36 0.312 -1.20 -0.67 -1.03 3.76 4.09 3.98 230 70 7 7 -0.999 -0.640 3.35 0.305 -1.30 -0.81 -1.14 3.88 4.19 4.08 230 80 7 7 -0.998 -0.680 3.34 0.298 -1.37 -0.93 -1.23 3.98 4.29 4.18 230 90 7 7 -0.998 -0.751 3.29 0.286 -1.43 -1.04 -1.31 4.08 4.38 4.28 230 100 7 7 -0.997 -0.833 3.22 0.273 -1.48 -1.13 -1.37 4.17 4.47 4.37 231 50 7 5 -1.000 -1.278 2.20 0.219 -1.36 -1.19 -1.29 3.72 3.93 3.85 231 60 7 5 -1.000 -1.283 2.34 0.219 -1.37 -1.19 -1.30 3.87 4.09 4.00 231 70 7 5 -1.000 -1.274 2.48 0.220 -1.37 -1.20 -1.30 4.01 4.23 4.14 231 80 7 5 -1.000 -1.274 2.59 0.220 -1.38 -1.21 -1.31 4.12 4.34 4.25 231 90 7 5 -1.000 -1.278 2.68 0.220 -1.38 -1.21 -1.31 4.22 4.44 4.35 231 100 7 5 -1.000 -1.281 2.76 0.219 -1.39 -1.22 -1.32 4.32 4.54 4.45 232 40 7 7 -1.000 -0.937 3.18 0.258 -0.88 -0.37 -0.68 3.52 4.00 3.82 232 50 7 7 -1.000 -0.937 3.18 0.258 -1.05 -0.54 -0.85 3.68 4.16 3,98 232 60 7 7 -1.000 -0.927 3.19 0.260 -1.19 -0.68 -0.99 3.81 4.29 4.11 232 70 7 7 -1.000 -0.939 3.18 0.258 -1.30 -0.80 -1.11 3.92 4.40 4.22 232 80 7 7 -1.000 -0.944 3.17 0.257 -1.41 -0.90 -1 .21 4.02 4.50 4.31 232 90 7 7 -1.000 -0.968 3.14 0.254 -1.50 -1.00 -1.30 4.09 4.58 4.39 232 100 7 7 -1.000 -0.961 3.15 0.255 -1 .57 -1 .07 -1.38 4.17 4.66 4.47 233 50 7 7 -1.000 -0.569 3.32 0.319 -1.02 -0.45 -0.84 3.58 3.90 3.79 2:33 60 7 7 -0.999 -0.615 3.33 0.310 -1.10 -0.55 -0.93 3.67 4.01 3.90 233 70 7 7 -0.999 -0.664 3.33 0.301 -1.17 -0.63 -1.00 3.76 4.11 4.00 233 80 7 7 -0.999 -0.723 3.32 0.290 -1.22 -0.70 -1.05 3.83 4.21 4.08 233 90 7 7 -0.998 -0,802 3.29 0.278 -1.25 -0.75 -1.10 3.90 4.30 4.17 233 100 7 7 -0.996 -0.898 3.23 0.263 -1.27 -0.80 -1.13 3.97 4.39 4.25 234 70 6 5 -0.989 -0.441 3.68 0.347 -1.43 -1 .05 -1.28 4.14 4.31 4.25 234 84 9 7 -0.963 -0.246 4.10 0.401 -1 .54 -1 .07 -1.36 4.37 4.48 4.43 234 99 9 7 -0.962 -0.449 3.91 0.345 -1.57 -1.17 -1.41 4.44 4.60 4.54 235 40 18 18 -0.995 -0.888 2.97 0.265 -1.14 -0.28 -0.85 3.17 3.94 3.72 235 50 18 18 -0.999 -0.804 3.26 0.277 -1.17 -0.31 -0.89 3.50 4.19 3.97 235 60 18 18 -0.999 -0.813 3.41 0.276 -1 .21 -0.35 -0.93 3.69 4.37 4.16 235 70 18 18 -0.999 -0.829 3.49 0.273 -1.25 -0.40 -0.97 3.81 4.51 4.29 235 80 18 18 -0.999 -0.854 3.52 0.270 -1.31 -0.45 -1 .02 3.87 4.61 4.39 235 90 18 18 -0.998 -0.882 3.51 0.266 -1.36 -0.51 -1.08 3.93 4.69 4.46 235 100 18 18 -0.998 -0.897 3.50 0.264 -1.42 -0.57 -1.14 3.97 4.75 4.52 235 110 18 18 -0.997 -0.881 3.50 0.266 -1.49 -0.62 -1.20 4.02 4.78 4.56 236 40 10 9 -0.996 -0.772 3.29 0.282 -0.99 -0.09 -0.70 3.40 4.09 3.84 236 50 10 9 -0.996 -0.781 3.32 0.281 -1.10 -0.21 -0.81 3.52 4.21 3.96 236 60 10 9 -0.996 -0.780 3.34 0.281 -1.21 -0.32 -0.93 3.62 4.31 4.06 236 70 10 9 -0.996 -0.779 3.34 0.281 -1.32 -0.43 -1 .04 3.70 4.39 4.14 236 80 10 10 -0.997 -0.833 3.26 0.273 -1.44 -0.22 -1.06 3.40 4.46 4.13 237 1 28 12 -0.988 -0.620 3.83 0.309 -0.81 -0.40 -0.55 4.07 4.31 4.17 237 2 28 8 -0.996 -0.702 3.70 0.294 -0.86 -0.61 -0.70 4.12 4.30 4.19 237 3 28 8 -0.994 -0.922 3.45 0.260 -0.91 -0.70 -0.77 4.09 4.28 4.17 225 Table B.2. (continued) No. of Pointsc Id. PCA Thinning Linee Log Nf Log Bf Codea Cond.b nT n rd i3 a Pest Min. Max. Mean Min. Max. Mean 237 4 28 8 -0.995 -1.158 3.18 0.232 -0.95 -0.77 -0.84 4.07 4.27 4.15 237 5 28 5 -0.999 -1.280 3.01 0.219 -0.97 -0.86 -0.91 4.11 4.26 4.18 238 40 8 5 -0.997 -0.598 3.77 0.313 -0.54 -0.27 -0.44 3.93 4.10 4.03 238 50 8 5 -0.998 -0.591 3.80 0.314 -0.71 -0.43 -0.61 4.06 4.22 4.16 238 60 8 5 -0.998 -0.588 3.81 0.315 -0.86 -0.58 -0.76 4.16 4.32 4.26 238 70 8 5 -0.998 -0.589 3.81 0.315 -1.01 -0.73 -0.90 4.24 4.40 4.34 239 40 8 5 -0.999 -0.524 3.72 0.328 -0.71 -0.35 -0.58 3.90 4.10 4.02 239 50 8 5 -0.999 -0.524 3.78 0.328 -0.82 -0.45 -0.69 4.02 4.22 4.14 239 60 8 5 -0.998 -0.521 . 3.82 0.329 -0.94 -0.57 -0.80 4.11 4.31 4.23 239 70 8 5 -0.998 -0.524 3.82 0.328 -1.07 -0.71 -0.94 4.19 4.39 4.31 241 20 6 5 -0.990 -4.100 2.03 0.098 -0.25 -0.20 -0.23 2.84 3.08 2.99 241 30 6 5 -0.997 -4.571 1.86 0.090 -0.40 -0.35 -0.38 3.47 3.69 3.61 241 40 6 5 -0.984 -3.830 1.92 0.104 -0.51 -0.45 -0.49 3.66 3.89 3.80 241 50 6 5 -0.991 -3.935 1.50 0.101 -0.62 -0.57 -0.60 3.73 3.96 3.87 241 60 6 5 -0.984 -4.467 0.68 0.091 -0.74 -0.69 -0.72 3.77 4.00 3.91 242 56 6 3 -0.963 -0.393 3.23 0.359 -1.33 -1.29 -1.31 3.74 3.75 3.74 242 73 7 4 -0.964 -0.148 3.78 0.435 -1.34 -1.24 -1.28 3.97 3.98 3.97 242 90 8 4 -0.951 -0.119 4.08 0.447 -1.38 -1.22 -1.30 4.22 4.24 4.23 243 1 12 7 -0.998 -0.551 2.69 0.322 -2.33 -1.89 -2.13 3.73 3.96 3.86 243 2 7 5 -0.969 -0.480 3.38 0.338 -1.69 -1.20 -1.46 3.96 4.17 4.08 244 70 6 3 -1.000 -1.352 2.58 0.213 -1.31 -1.25 -1.28 4.27 4.35 4.31 244 86 6 4 -1.000 -0.898 3.24 0.263 -1.38 -1.25 -1.32 4.36 4.48 4.42 244 fo2 6 5 -0.996 -0.753 3.49 0.285 -1.44 -1.23 -1.34 4.41 4.57 4.51 245 1 12 5 -0.995 -0.568 3.28 0.319 -1.63 -1.45 -1.54 4.10 4.20 4.15 245 2 11 5 -0.991 -0.437 3.54 0.348 -1.73 -1.53 -1 .63 4.21 4.29 4.25 245 3 9 4 -0.998 -0.446 3.55 0.346 -1.79 -1.63 -1.72 4.28 4.35 4.32 246 30 4 4 -0.997 -0.874 3.08 0.267 -0.56 0.01 -0.29 3.07 3.56 3.34 246 50 6 4 -0.984 -0.470 3.51 0.340 -1.01 -0.49 -0.75 3.72 3.96 3.86 246 60 7 4 -0.998 -0.352 3.74 0.370 -1.24 ..:o.8o -1.04 4.02 4.17 4.11 246 70 7 4 -0.992 -0.348 3.81 0.371 -1.35 -0.91 -1.15 4.12 4.27 4.21 246 80 7 4 -0.997 -0.361 3.83 0.367 -1.44 -1 .01 -1.24 4.19 4.34 4.28 247 1 7 4 -0.990 -3.804 2.06 0.104 -0.57 -0.55 -0.56 4.13 4.22 4.18 247 2 7 4 -0.988 -4.973 1.53 0.084 -0.58 -0.56 -0.57 4.31 4.43 4.38 247 3 7 4 -0.994 -6.286 0.85 0.069 -0.60 -0.58 -0.59 4.47 4.61 4.55 247 4 7 4 -0.991 -8.132 -0.21 0.055 -0.62 -0.59 -0.60 4.60 4.79 4.70 248 1 9 5 -0.996 -6.444 -3.17 0.067 -1.07 -0.95 -1.02 2.98 3.69 3.40 248 2 10 6 -0.999 -6.855 -3.66 0.064 -1.12 -0.97 -1.06 3.02 4.01 3.59 248 3 11 7 -0.992 -7.224 -4.18 0.061 -1.16 -0.98 -1.09 2.98 4.22 3.72 249 100 9 3 -0.995 -1.243 2.53 0.223 -1.36 -1.27 -1.31 4.11 4.22 4.17 249 110 9 3 -0.998 -1.188 2.64 0.229 -1.39 -1.30 -1.34 4.18 4.29 4.24 249 120 9 3 -0.999 -1.015 2.88 0.248 -1.44 -1.33 -1.38 4.23 4.34 4.29 250 2 13 4 -0.993 -0.659 3.69 0.301 -1.18 -1.09 -1.13 4.41 4.47 4.44 250 3 13 8 -0.993 -0.549 3.82 0.323 -1 .39 -1.10 -1.26 4.42 4.59 4.51 251 1 27 27 -0.999 -0.645 3.74 0.304 -1.77 -0.42 -1.30 3.99 4.87 4.58 226 Table B.2. (continued) No. of Log Nf Log Bf Pointsc PCA Thinning Linee Id. Codea Cond.b nT n rd s & Pest Min. Max. Mean Min. Max. Mean 252 120 9 3 -0.993 -4.466 -1.62 0.091 -1.40 -1.36 -1.38 4.47 4.62 4.55 252 140 9 5 -0.987 -4.593 -1.78 0.089 -1.42 -1.36 -1.39 4.43 4.74 4.60 252 160 9 9 -0.996 -4.748 -1.93 0.087 -1.43 -1.21 -1.35 3.82 4.84 4.48 252 180 9 9 -0.999 -3.084 0.56 o. 122 -1.43 -1.13 -1.32 4.07 4.94 4.63 252 200 9 9 -0.999 -2.392 1.66 0.147 -1.41 -1.07 -1.29 4.24 5.03 4.75 252 220 9 9 -0.999 -2.043 2.25 0.164 -1.39 -1.03 -1.27 4.36 5.11 4.84 252 240 9 9 -0.999 -1.814 2.67 0.178 -1.38 -0.99 -1.24 4.46 5.17 4.92 253 47 14 10 -0.999 -0.858 3.35 0.269 -1.18 -0.59 -0.93 3.85 4.35 4.14 253 57 15 13 -0.998 -0.727 3.49 0.290 -1.42 -0.39 -1.00 3.77 4.48 4.22 253 66 15 14 -0.996 -0.618 3.60 0.309 -1.65 -0.36 -1.15 3.82 4.59 4.31 254 50 12 12 -1.000 -0.947 3.24 0.257 -1.25 -0.57 -1.00 3.77 4.42 4.18 254 61 13 9 -0.999 -0.691 3.64 0.296 -1.34 -1.00 -1.19 4.32 4.56 4.46 254 70 13 11 -0.992 -0.809 3.51 0.276 -1.44 -1.03 -1.28 4.32 4.66 4.55 255 87 14 7 -0.983 -0.547 4.39 0.323 -1.76 -1.26 -1.54 5.06 5.33 5.23 256 40 7 4 -0.997 -0.868 2.91 0.268 -0.90 -0.51 -0.72 3.35 3.68 3.54 256 53 7 4 -0.995 -0.773 3.03 0.282 -1.08 -0.67 -0.89 3.53 3.85 3. 72 256 66 7 4 -0.992 -0.683 3.14 0.297 -1.20 -0.78 -1.01 3.66 3.94 3.83 257 45 13 13 -1.000 -0.541 3.07 0.324 -1.28 -0.10 -0.95 3.13 3.76 3.58 257 49 13 11 -0.999 -0.655 3.00 0.302 -1.20 -0.55 -0.97 3.37 3.79 3.64 257 53 1311 -1.000 -0.874 3.04 0.267 -1.08 -0.62 -0.91 3.59 3.99 3.84 257 54 13 11 -0.999 -1.011 2.98 0.249 -1.03 -0.55 -0.85 3.55 4.03 3.84 257 57 13 10 -1.000 -1.731 2.51 o. 183 -0.91 -0.74 -0.84 3.79 4.09 3.97 258 40 9 6 -0.991 -0.733 3.26 0.289 -1.19 -0.87 -1.09 3.89 4.12 4.06 258 52 10 6 -0.998 -0.824 3.19 0.274 -1.35 -1.04 -1.24 4.05 4.31 4.22 258 60 10 6 -0.997 -0.711 3.40 0.292 -1.48 -1.17 -1.37 4.23 4.45 4.37 259 100 17 14 -0.994 -0.466 4.05 0.341 -1.59 -0.80 -1.27 4.40 4.78 4.65 259 110 17 13 -0.998 -0.425 4.14 0.351 -1.64 -0.98 -1.36 4.54 4.83 4.71 259 120 17 13 -0.996 -0.407 4.19 0.355 -1.69 -1.03 -1.40 4.59 4.86 4.76 259 130 17 13 -0.997 -0.396 4.23 0.358 -1.72 -1.08 -1.44 4.64 4.90 4.80 259 140 17 13 -0.999 -0.390 4.25 0.360 -1.75 -1.11 -1.47 4.68 4.93 4.83 259 150 17 13 -0.999 -0.378 4.29 0.363 -1.79 -1.14 -1 .51 4.71 4.97 4.86 259 160 17 13 -0.999 -0.379 4.31 0.363 -1.81 -1.16 -1.53 4.74 4.99 4.89 259 170 17 13 -0.998 -0.383 4.32 0.362 -1.83 -1.18 -1.55 4.77 5.02 4.92 259 180 17 13 -0.999 -0.376 4.35 0.363 -1.84 -1.20 -1.57 4.80 5.05 4.94 259 190 17 13 -0.999 -0.377 4.37 0.363 -1.86 -1.22 -1.59 4.82 5.07 4.97 259 200 17 13 -0.999 -0.377 4.39 0.363 -1.87 -1.24 -1.60 4.85 5.10 4.99 259 210 17 13 -1.000 -0.375 4.40 0.364 -1.89 -1.25 -1.62 4.87 5.11 5.01 260 60 14 7 -0.999 -0.563 4.00 0.320 -0.85 -0.59 -0.72 4.33 4.48 4.41 260 70 14 9 -0.999 -0.563 4.03 0.320 -0.96 -0.59 -0.79 4.36 4.57 4.47 260 80 15 9 -0.997 -0.555 4.04 0.322 -1.08 -0.73 -0.91 4.44 4.64 4.55 260 90 15 9 -0.998 -0.513 4.11 0.331 -1.18 -0.79 -0.99 4.51 4.71 4.62 260 100 15 9 -0.998 -0.480 4.15 0.338 -1.26 -0.88 -1.08 4.57 4.75 4.67 260 110 15 9 -0.995 -0.455 4.20 0.344 -1.34 -0.95 -1.15 4.63 4.80 4.73 260 120 15 9 -0.997 -0.411 4.26 0.354 -1.41 -1.03 -1.23 4.68 4.84 4.77 227 ? Table B.2. (continued) , .. No. of Log Nf Log Bf Pointsc PCA Thinning Linee I d. Code a Cond.b nT n rd s &. Pest Min. Max. Mean Min. Max. Mean 260 130 15 9 -0.998 -0.402 4.29 0.357 -1.46 -1.07 -1.27 4.72 4.88 4.80 260 140 15 9 -0.999 -0.382 4.33 0.362 -1.51 -1.13 -1.33 4.76 4.90 4.84 260 150 15 9 -0.999 -0.377 4.35 0.363 -1 .55 -1.15 -1.36 4.78 4.93 4.86 260 160 15 9 -0.999 -0.370 4.37 0.365 -1.57 -1.18 -1.39 4.81 4.96 4.89 260 170 15 9 -0.994 -0.383 4.37 0.362 -1 .59 -1.25 -1.42 4.83 4.98 4.91 260 180 15 9 -1.000 -0.358 4.42 0.368 -1.62 -1.23 -1.43 4.85 5.00 4.93 261 60 14 8 -0.999 -0.612 3.93 0.310 -0.85 -0.53 -0.70 4.26 4.46 4.36 261 70 14 8 -0.999 -0.597 3.97 0.313 -0.96 -0.65 -0.82 4.35 4.55 4.45 261 80 15 9 -0.997 -0.608 3.97 0.311 -1.08 -0.73 -0.91 4.40 4.62 4.52 261 90 15 9 -0.999 -0.543 4.05 0.324 -1.18 -0.79 -0.99 4.47 4.68 4.59 261 100 15 9 -0.997 -0.528 4.08 0.327 -1.26 -0.88 -1.08 4.53 4.74 4.65 261 110 15 9 -0.997 -0.499 4.12 0.333 -1.34 -0.95 -1.15 4.59 4.78 4.70 261 120 15 9 -0.997 -0.476 4.16 0.339 -1.41 -1.03 -1.23 4.64 4.83 4.74 261 130 15 9 -0.997 -0.440 4.22 0.347 -1.46 -1.07 -1.27 4.68 4.86 4.78 261 140 15 9 -0.995 -0.423 4.25 0.351 -1.51 -1.13 -1.33 4.72 4.89 4.82 261 150 15 9 -0.998 -0.414 4.28 0.354 -1.55 -1.15 -1.36 4.75 4.92 4.84 261 160 15 9 -0.999 -0.410 4.30 0.355 -1 .57 -1.18 -1.39 4.78 4.94 4.87 ? 262 70 13 13 -0.998 -0.584 4.02 0.316 -0.85 0.34 -0.43 3.80 4.49 4.27 262 80 13 13 -0.998 -0.580 4.01 0.316 -0.98 0.23 -0.56 3.86 4.56 4.34 262 90 13 13 -0.997 -0.595 3.99 0.314 -1.09 0.07 -0.68 3.92 4.62 4.39 262 100 13 13 -0.997 -0.595 3.99 0.313 -1 .21 -0.04 -0.79 3.98 4.68 4.46 262 110 13 13 -0.998 -0.598 3.98 0.313 -1.31 -0.14 -0.89 4.04 4.74 4.52 262 120 13 13 -0.998 -0.598 3.98 0.313 -1.38 -0.23 -0.98 4.10 4.80 4.57 262 130 13 13 -0.998 -0.600 3.99 0.313 -1.46 -0.32 -1.07 4.15 4.85 4.63 262 140 13 13 -0.998 -0.599 3.99 0.313 -1.55 -0.39 -1.15 4.20 4.90 4.67 262 150 13 13 -0.998 -0.587 4.00 0.315 -1.63 -0.47 -1.23 4.25 4.94 4. 72 263 26 9 4 -0.968 -0.848 3.72 0.271 -0.40 -0.31 -0.36 3.98 4.05 4.02 263 33 11 7 -0.998 -0.850 3.72 0.270 -0.60 -0.32 -0.48 3.98 4.22 4.12 263 39 11 9 -0.996 -0.776 3.72 0.281 -0.77 -0.32 -0.59 3.96 4.30 4.18 264 40 9 4 -0.936 -0.243 3.51 0.402 -0.94 -0.56 -0.72 3.63 3.73 3.68 264 50 11 7 -0.938 -0.551 3.41 0.322 -0.94 -0.60 -0.76 3.71 3.90 3.83 265 77 6 4 -0.979 -0.250 4.06 0.400 -1.38 -0.90 -1.15 4.28 4.39 4.35 266 60 12 8 -0.995 -0.256 4.16 0.398 -1.31 -0.96 -1.16 4.40 4.49 4.45 266 80 14 11 -0.989 -0.336 4.14 0.374 -1.55 -0.93 -1.29 4.44 4.65 4.58 266 100 14 11 -0.993 -0.334 4.22 0.375 -1 .73 -1.13 -1.48 4.58 4.79 4. 71 266 120 14 12 -0.993 -0.331 4.26 0.376 -1.87 -1.16 -1.58 4.62 4.87 4.78 266 140 8 8 -0.964 -0.566 3.93 0.319 -1.68 -1.01 -1.37 4.44 4.84 4.71 267 42 12 8 -0.987 -0.719 3.62 0.291 -0.96 -0.63 -0.79 4.06 4.29 4.19 268 54 12 7 -0.996 -0.381 3.71 0.362 -1 .02 -0.65 -0.86 3.95 4.10 4.04 269 75 6 6 -0.994 -0.654 3.43 0.302 -1.57 -0.42 -1.09 3.68 4.42 4.15 270 60 14 14 -0.994 -0.597 3.68 0.313 -1.17 -0.22 -0.85 3.79 4.34 4.19 270 70 14 14 -0.994 -0.626 3.64 0.308 -1.30 -0.34 -0.98 3.83 4.43 4.26 270 80 14 14 -0.994 -0.627 3.65 0.307 -1.40 -0.45 -1.08 3.91 4.50 4.33 270 90 14 14 -0.991 -0.625 3.67 0.308 -1.48 -0.52 -1.16 3.97 4.56 4.40 228 Table 8.2. (continued) No. of Pointsc Id. PCA Thinning Linee Log Nf Log Bf Codea Cond.b nT n rd 8 & Pest Min. Max. Mean Min. Max. Mean 270 100 14 14 -0.990 -0.641 3.68 0.305 -1 .55 -0.59 -1.22 4.02 4.63 4.46 270 110 14 14 -0.991 -0.656 3.69 0.302 -1.59 -0.64 -1.26 4.08 4.73 4.52 270 120 14 14 -0.991 -0.646 3.73 0.304 -1.63 -0.68 -1.31 4.13 4.74 4.58 271 40 18 18 -0.995 -0.888 2.97 0.265 -1.14 -0.28 -0.85 3.17 3.94 3.72 271 50 18 18 -0.999 -0.804 3.26 0.277 -1.17 -0.31 -0.89 3.50 4.19 3.97 271 60 18 18 -0.999 -0.813 3.41 0.276 -1.21 -0.35 -0.93 3.69 4.37 4.16 271 70 18 18 -0.999 -0.829 3.49 0.273 -1.25 -0.40 -0.97 3.81 4.51 4.29 271 80 18 18 -0.999 -0.854 3.52 0.270 -1.31 -0.45 -1.02 3.87 4.61 4.39 271 90 18 18 -0.998 -0.882 3.51 0.266 -1.36 -0.51 -1.08 3.93 4.69 4.46 271 100 18 18 -0.998 -0.897 3.50 0.264 -1.42 -0.57 -1.14 3.97 4.75 4.52 271 110 18 18 -0.997 -0.881 3.50 0.266 -1.49 -0.62 -1.20 4.02 4.78 4.56 272 50 8 6 -0.998 -1.656 2.12 0.188 -1.25 -1.08 -1.18 3.90 4.18 4.07 272 60 9 7 -0.956 -1.855 1.89 0.175 -1.36 -1.14 -1.25 3.94 4.35 4.21 272 71 9 7 -0.984 -2.373 1.18 0.148 -1.41 -1.25 -1.34 4.11 4.50 4.37 272 81 10 8 -0.997 -2.439 1.08 0.145 -1.44 -1.26 -1.38 4.15 4.61 4.45 272 103 10 10 -0.999 -1.848 1.93 0.176 -1.49 -1.06 -1.37 3.88 4.69 4.46 273 40 18 15 -0.995 -0.404 3.85 0.356 -0.97 -0.06 -0.67 3.87 4.22 4.12 273 50 18 15 -0.988 -0.437 3.90 0.348 -1.11 -0.34 -0.82 4.02 4.36 4.26 273 60 18 15 -0.990 -0.449 3.94 0.345 -1.24 -0.47 -0.95 4.13 4.47 4.37 273 70 18 15 -0.993 -0.454 3.98 0.344 -1.34 -0.58 -1.06 4.22 4.57 4.46 273 80 18 15 -0.993 -0.414 4.05 0.354 -1.45 -0.70 -1.18 4.36 4.64 4.54 273 90 18 15 -0.994 -0.481 3.96 0.338 -1.59 -0.84 -1.31 4.34 4.70 4.59 273 100 18 15 -0.988 -0.471 3.95 0.340 -1.76 -1.00 -1.49 4.39 4.76 4.65 274 60 10 8 -0.999 -0.378 3.96 0.363 -0.94 -0.42 -0.74 4.12 4.32 4.24 274 70 10 8 -0.997 -0.399 3.98 0.357 -1.08 -0.56 -0.89 4.20 4.41 4.33 274 80 10 8 -0.998 -0.403 3.99 0.356 -1.21 -0.69 -1.01 4.27 4.48 4.40 274 90 10 9 -0.998 -0.422 3.97 0.352 -1.32 -0.69 -1.08 4.26 4.54 4.43 274 100 10 9 -0.998 -0.440 3.95 0.347 -1.43 -0.81 -1.19 4.30 4.58 4.47 276 84 9 9 -0.998 -1.222 3.35 0.225 -1 .06 -0.60 -0.88 4.09 4.62 4.43 276 95 9 9 -0.997 -1.220 3.42 0.225 -1.07 -0.62 -0.89 4.17 4.70 4.51 276 112 9 9 -0.998 -1.224 3.44 0.225 -1.10 -0.65 -0.93 4.24 4.77 4.58 277 70 5 4 -1.000 -0.333 3.74 0.375 -1.05 -0.36 -0.75 3.86 4.09 3.98 277 80 9 7 -0.996 -0.362 3.79 0.367 -1.40 -0.52 -1.09 3.99 4.31 4.19 277 90 10 9 -0.995 -0.396 3.83 0.358 -1.50 -0.27 -1.12 3.94 4.45 4.27 277 100 10 9 -0.993 -0.414 3.88 0.354 -1 .54 -0.40 -1.20 4.06 4.54 4.38 277 110 10 9 -0.990 -0.418 3.95 0.353 -1.55 -0.46 -1.24 4.16 4.63 4.46 277 120 8 7 -0.994 -0.395 4.03 0.358 -1 .51 -0.50 -1.18 4.24 4.65 4.50 278 1 33 19 -1.000 -0.373 4.31 0.364 -1.92 -1.29 -1.60 4.79 5.02 4.90 279 40 11 10 -0.993 -0.640 3.72 0.305 -1.53 -0.52 -1.14 4.02 4.67 4.45 280 51 8 6 -0.988 -0.552 3.83 0.322 -1.71 -1.41 -1.57 4.60 4.76 4.70 281 27 5 5 -0.992 -0.691 4.09 0.296 -0.89 -0.17 -0.57 4.18 4.67 4.48 281 30 5 5 -0.990 -0.621 4.11 0.308 -1.00 -0.29 -0.68 4.27 4.70 4.53 281 33 5 5 -0.979 -0.547 4.14 0.323 -1.11 -0.41 -0.79 4.34 4.71 4.58 Table B.2. (continued) I d. No. of Pointsc 229 PCA Thinning Linee Log Bf Codea Cond.b nr n " a Pest Min. Max. Mean Min. Max. Mean 281 27 5 5 -o.gg2 -o.6gl 4.09 0.296 -0.89 -0.17 -0.57 4.18 4.67 4.48 281 30 5 5 -0.990 -0.621 4.11 0.308 -1.00 -0.29 -0.68 4.27 4. 70 4.53 281 33 5 5 -0.979 -0.547 4.14 0.323 -1.11 -0.41 -0.79 4.34 4. 71 4.58 arable B.l associates each Id. code with a particular yield table. bsee Table B. 1 and the references given there for more information on condition. CnT is the total number of log B-log N points given for each code and condition. n is the number of points remaining after removing points not relevant to the thinning line. This is the number of points used to fit th.e PCA relationship between log B and log N. drhese correlation coefficients have little statistical meaning because the natural variability present in real forest measurements has been removed from yield table predictions. These values are included only as a crude index of variation around the fitted line. eg and~ are, respectively, the slope and intercept of the fitted PCA thinning line. Formulas for? principle components analysis are given in Jolicoeur and H~usner {1971). Pest is a transformation of the thinning slope ~alculated by Pest= 0.5 I {1- E). See discussion in Chapter 6. fThe mean, minimum, and maximum are given for log B and for log N over the n data points used to fit each thinning line. 230 Table 8.3. Fitted Allometric Relationships for Forestry Yield Table Data. Allometric Relationships Fit bl Princieal Component Anallsisc 109 fi - lo9 w lo9 OBH - lo9 w log BSLA - 109 w log ii - log OBH 109 ii - log BSLA !d. " " " " " Code a Cond .b r ow r sw r <~>ho n r hB 201 40 13 0.988 0.159 13 1.000 0.381 13 1.000 0.764 13 0.985 0.417 13 0.986 0.207 201 50 14 0.994 0.188 14 0.998 0.381 14 1.000 0.763 14 0.993 0.492 14 0.993 0.246 201 60 14 0.997 0.215 14 1.000 0.381 14 1.000 0.755 14 0.998 0.564 14 0.997 0.284 201 70 14 1.000 0.234 14 1.000 0.377 14 1.000 0.754 14 1.000 0.620 14 0.999 0.310 201 80 14 1.000 0.250 14 1.000 0.379 14 1.000 0.757 14 1.000 0.660 14 1.000 0.331 201 90 14 1.000 0.257 14 1.000 0.379 14 1.000 0.759 14 1.000 0.679 14 1.000 0.339 201 100 14 0.999 0.271 14 1.000 0.381 14 1.000 0.761 14 0.999 0. 7ll 14 0.999 0.355 201 llO 14 0.999 0.278 14 0.999 0.384 14 1.000 0.765 14 0.999 0.724 14 0.999 0.363 201 120 14 1.000 0.281 14 1.000 0.380 14 1 .000 0.759 14 0.999 0.738 14 0.999 0.370 201 130 14 1.000 0.280 14 0.999 0.380 14 0.999 0. 758 14 0.999 o. 736 14 0.999 0.369 201 140 4 0.999 0.273 4 1.000 0.333 4 1.000 0.668 4 1.000 0.821 4 1.000 0.409 201 150 4 0.999 0.275 4 1.000 0.339 4 1.000 0.676 4 1.000 0.810 4 1.000 0.406 201 160 4 1.000 0.271 4 1.000 0.326 4 1.000 0.656 4 1.000 0.832 4 1.000 0.414 202 80 9 0.999 0.231 9 1.000 0.347 9 0.998 0.665 202 96 9 0.999 0.229 9 1.000 0.392 9 0.999 0.584 202 ll2 9 1.000 0.2ll 9 1.000 0.393 9 0.999 0.537 203 30 8 0.995 0.215 8 1.000 0.322 8 1.000 0.662 8 0.994 0.670 8 0.994 0.325 203 40 8 0.998 0.208 8 1.000 0.332 8 1.000 0.653 8 0.997 0.626 8 0.997 0.318 203 50 8 0.993 0.219 8 1.000 0.322 8 1.000 0.656 8 0.991 0.681 8 0.991 0.333 203 60 8 0.997 0.213 8 1.000 0.330 8 1.000 0.664 8 0.995 0 .? 644 8 0.996 0.320 203 70 8 0.996 0.209 8 1.000 0.332 8 1.000 0.669 8 0.995 0.630 8 0.995 0.313 203 80 8 0.998 0.216 8 1.000 0.327 8 1.000 0.660 8 0.998 0.661 8 0.999 0.328 203 90 8 0.999 0.222 8 1.000 0.329 8 1.000 0.660 8 0.999 0.675 8 0.999 0.336 204 62 7 0.997 0.205 7 0.999 0.427 7 1.000 0.849 7 0.995 0.480 7 0.996 0.241 204 71 8 0.995 0.189 8 1.000 0.408 8 1.000 0.818 8 0.993 0.462 8 0.994 0.230 204 80 8 0.997 0.174 8 1.000 0.404 8 1.000 0.818 8 0.997 0.431 8 0.997 0.213 205 58 8 0.998 0.250 8 1.000 0.444 8 1.000 0.884 8 0.998 0.563 8 0.998 0.282 205 68 8 0.996 0.234 8 1.000 0.455 8 1.000 0.915 8 0.996 0.515 8 0.996 0.256 205 80 8 0.997 0.192 8 1.000 0.449 8 1.000 0.898 8 0.997 0.429 8 0.997 0.214 206 52 9 0.999 0.264 9 0.999 0.369 9 1.000 0.730 9 0.998 0. 715 9 0.998 0.362 206 64 9 0.994 0.233 9 1.000 0.408 9 1.000 0.824 9 0.995 0.572 9 0.994 0.283 206 75 9 0.998 0.217 9 1.000 0.428 9 1.000 0.857 9 0.998 0.508 9 0.998 0.254 207 49 ll 0.995 0.258 ll 0.993 0.333 11 0.982 0.770 208 40 6 0.999 0.301 6 1.000 0.375 6 1.000 0.748 6 0.999 0.804 6 0.999 0.402 208 50 6 0.999 0.302 6 1.000 0.372 6 1.000 0.745 6 0.999 0.812 6 0.999 0.405 208 60 6 1.000 0.302 6 1.000 0.376 6 1.000 0.745 6 1.000 0.802 6 0.999 0.405 208 70 6 0.999 0.294 6 1.000 0.372 6 1.000 0.754 6 0.999 0.790 6 0.999 0.390 209 1 4 0.997 0.166 4 0.999 0.837 209 2 6 0.988 0.202 6 0.999 0.862 209 3 0.965 0. ll8 7 0.992 0.251 7 0.999 0.889 0.988 0.479 0.953 0.131 209 4 8 0.998 0.433 8 0.999 0.844 210 136 28 0.998 0.166 28 0.998 0.392 28 0.999 0.422 211 20 14 1.000 0.217 14 0.999 0.406 14 1.000 0.780 14 1.000 0.535 14 1.000 0.278 231 Table B.3. (continued) Allometric Relationships Fit b,l Princieal Component Anal ;ts is c log ii - log w log DBH - log w log BSLA - log w log ii - log DBH log ii - log BSLA ld. A A A A A Codea Cond .b r ~hw <~>ow <~>sw r hO r hB 211 30 14 0.999 0.220 14 0.999 0.399 14 1.000 0.790 14 0.999 0.553 14 1.000 0.279 211 40 14 1.000 0.217 14 1.000 0.402 14 1.000 0.800 14 1.000 0.538 14 1.000 0.271 211 50 14 1.000 0.219 14 1.000 0.397 14 1.000 0.792 14 1.000 0.552 14 1.000 0.277 211 60 14 1.000 0.218 14 1.000 0.400 14 1.000 0. 792 14 1.000 0.545 14 1.000 0.276 211 70 14 1.000 0.217 14 1.000 0.400 14 1.000 0.793 14 1.000 0.544 14 1.000 0.274 212 60 14 0.999 0.278 14 1.000 0.397 14 1.000 0.790 14 0.999 0. 701 14 0.999 0.352 212 80 15 1.000 0.282 15 0.999 0.387 15 1.000 0.777 15 0.999 0.728 15 0.999 0.363 212 100 15 1.000 0.279 15 1.000 0.391 15 1.000 0.781 15 0.999 0.714 15 0.999 0.357 212 120 15 1.000 0.282 15 1.000 0.390 15 1.000 0.778 15 0.999 0.722 15 0.999 0.362 212 140 15 1.000 0.286 15 1.000 0.386 15 1.000 o. 773 15 0.999 0. 742 15 0.999 0.371 212 160 15 1.000 0.283 15 1.000 0.389 15 1.000 0.778 15 0.999 0.729 15 0.999 0.364 212 180 15 1.000 0.292 15 1.000 0.384 15 1.000 0.768 15 0.999 0.760 15 0.999 0.380 212 200 15 1.000 0.284 15 1.000 0.387 15 1.000 0.774 15 0.999 0.735 15 0.999 0.367 213 30 6 0.997 0.260 6 0.999 0.404 6 1.000 0. 795 6 0.993 0.644 6 0.997 0.328 213 40 6 0.994 0.291 6 1.000 0.396 6 1.000 0.799 6 0.994 0.738 6 0.994 0.365 213 50 6 0.996 0.288 6 0.999 0.394 6 1.000 0.796 6 0.996 0. 731 6 0.996 0.362 213 60 6 0.999 0.278 6 1.000 0.400 6 1.000 0.800 6 0.998 0.697 6 0.999 0.34B 213 70 6 1.000 0.271 6 1.000 0.396 6 1.000 0.797 6 1.000 0.685 6 1.000 0.340 214 1 214 2 214 3 214 4 214 5 214 6 215 l 215 2 215 3 215 4 215 5 215 6 216 l 216 2 216 3 217 54 6 0.997 0.242 6 1.000 0.372 6 1.000 0.745 6 0.995 0.651 6 0.995 0.325 217 64 6 0.998 0.266 6 1.000 0.391 6 1.000 0.781 6 0.996 0.682 6 0.996 0.341 217 75 6 0.999 0.280 6 1.000 0.408 6 0.999 0.814 6 0.998 0.687 6 0.997 0.343 218 50 10 0.999 0.360 10 1.000 0.336 10 1.000 0.673 10 0.998 1.072 10 0.998 0.535 218 60 10 1.000 0.350 10 1.000 0.330 10 1.000 0.662 10 1.000 1.060 10 1.000 0.529 218 70 10 1.000 0.346 10 1.000 0.339 10 1.000 0.669 10 0.999 1.022 10 0.999 0.517 220 40 6 0.999 0.353 6 0.999 0.292 6 0.999 0.579 6 0.999 1.212 6 1.000 0.611 220 50 6 0.998 0.368 6 0.999 0.286 6 1.000 0.575 6 0.998 1.288 6 0.998 0.641 220 60 6 0.997 0.366 6 1.000 0.297 6 1.000 0.584 6 0.997 1. 237 6 0.996 0.627 220 70 6 0.995 0.348 6 0.999 0.302 6 1.000 0.611 6 0.990 1.156 6 0.992 0.569 232 Table 8.3. (cant i nued) Allometric Relationships Fit b~ Principal Component Ana 1 ~s i sc log fi - log w log OBH - log w log BSLA - log w log h - log DBH log fi - log BSLA ld. 1\ " 1\ 1\ $hB Codea Cond .b hw ~Ow ~Bw r ~hO r 220 80 6 0.993 0.321 6 0.999 0.321 6 0.999 0.642 6 0.988 1.008 6 0.987 0.499 220 90 6 0.988 0.292 6 0. 999 0.340 6 0.999 0.674 6 o. 979 0.866 6 0.980 0.432 221 80 13 0.998 0.216 13 1.000 0.415 13 0.999 0.825 13 0.996 0.519 13 0.996 0.251 221 90 13 0.999 0.223 13 0.999 0.408 13 0.999 0.821 13 0.998 0.546 13 0.998 0.272 221 100 13 0.999 0.227 13 0.999 0.410 13 0.999 0.821 13 0.997 0.552 13 0.997 0.276 221 110 13 0.999 0.226 13 0. 999 0.406 13 0.999 0.814 13 0.997 0.556 13 0.997 0.277 221 120 13 0.999 0.231 13 0.999 0.410 13 0.999 0.817 13 0.997 0.563 13 0.997 0.282 221 130 13 0.999 0.231 13 0.999 0.407 13 0.999 0.813 13 0.996 0.566 13 0.996 0.283 221 140 13 0.998 0.232 13 0.999 0.410 13 0.999 0.815 13 0.996 0.565 13 0.996 0.284 221 150 13 0.998 0.233 13 1.000 0.409 13 0.999 0.813 13 0.996 0.569 13 0.996 0.286 221 160 13 0.999 0.234 13 0.999 0.408 13 0.999 0.814 13 0.996 0.572 13 0.996 0.286 221 170 13 0.998 0.233 13 0.999 0.406 13 0.999 0.815 13 0.995 0.572 13 0.995 0.285 221 180 13 0.998 0.233 13 1.000 0.404 13 0.999 0.813 13 0.996 0.576 13 0.996 0.285 221 190 13 0.999 0.235 13 0.999 0.406 13 0.999 0.814 13 0.996 0.578 13 0.996 0.288 221 200 13 0.998 0.235 13 0.999 0.406 13 0.999 0.812 13 0.996 0.577 13 0.996 0.288 221 210 13 0.998 0.233 13 0.999 0.407 13 0.999 0.816 13 0.996 0.572 13 0.995 0.285 222 1 8 0.999 0.324 8 0.999 0.415 8 1.000 0.843 8 0.999 0.780 8 1.000 0.384 222 2 8 1.000 0.349 8 0.999 0.401 8 1.000 0.817 8 0.999 0.872 8 1.000 0.428 222 3 8 1.000 0.383 8 1.000 0.395 8 1.000 0.801 8 0.999 0.969 8 1.000 0.478 222 4 5 1.000 0.404 5 0.999 0.392 5 1.000 0. 780 5 1.000 1.030 5 1.000 0.517 222 5 5 1.000 0.425 5 0.999 0.384 5 1 .000 0.749 5 0.999 1.106 5 1.000 0.567 223 1 13 0.998 0.223 13 1.000 0.423 13 0.998 0.527 224 3 14 0.997 0.193 14 0.993 0.311 14 0.994 0.618 225 1 6 0.994 0.248 6 1.000 0.404 6 0.993 0.616 227 40 13 0.999 0.306 13 0.999 0.337 13 0.999 0.662 13 0.998 0.909 13 0.997 0.462 227 50 13 1.000 0.312 13 1.000 0.340 13 0.999 0.663 13 0.999 0.917 13 0.998 0.470 227 60 13 0.999 0.309 13 1.000 0.339 13 0.999 0.663 13 0.999 0.911 13 0.998 0.465 227 70 13 0.999 0.308 13 1.000 0.337 13 0.999 0.663 13 0.999 0.914 13 0.998 0.465 227 80 13 0.999 0.310 13 1.000 0.336 13 0.999 0.663 13 0.999 0.922 13 0.998 0.468 227 90 13 0.999 0.310 13 1.000 0.341 13 0.999 0.662 13 o. 999 0.908 13 0.998 0.467 227 100 13 0.999 0.309 13 1.000 0.338 13 0.999 0.663 13 0.999 0.916 13 0.998 0.466 227 110 13 0.999 0.310 13 1.000 0.338 13 0.999 0.663 13 0.999 0.916 13 0.998 0.467 227 120 13 0.994 0.312 13 0.995 0.341 13 0.997 0.662 13 0.999 0.915 13 0.984 0.468 228 25 11 0.997 0.347 11 1.000 0.399 11 1.000 0.808 11 0.996 0.869 11 0.997 0.429 228 35 11 0.999 0.325 11 1.000 0.410 11 1.000 0.806 11 0.998 0. 791 11 0.999 0.403 228 45 12 0.999 0.312 12 1.000 0.404 12 1.000 0.809 12 o. 999 0.774 12 1.000 0.386 228 55 12 1.000 0.306 12 1.000 0.412 12 1.000 0.823 12 0.999 0.743 12 0.999 0.372 228 65 12 0.999 0.289 12 1.000 0.421 12 1.000 0.838 12 0.999 0.687 12 0.999 0.345 228 75 12 0.998 0.291 12 0.999 0.428 12 1.000 0.851 12 0.998 0.680 12 0.997 0.342 229 60 7 1.000 0.303 7 1.000 0.351 7 1.000 0.710 7 1.000 0.863 7 1.000 0.427 229 70 7 1.000 0.300 7 1.000 0.353 7 1.000 0.707 7 1.000 0.849 7 1.000 0.424 229 80 7 1.000 0.301 7 1.000 0.349 7 1.000 0.709 7 1.000 0.861 7 1.000 0.424 229 90 7 1.000 0.306 7 1.000 0.356 7 1.000 0.710 7 1 .000 0.861 7 1.000 0.431 233 Table 8.3. (continued) Allometric Relationships Fit b.)' Principal Component Anal.)'sisc log h - log w log DBH - log w log BSLA - log w log h - log DBH log fi - log BSLA ld. A A A A A Codea Cond .b ~hw sw hO hB 229 100 7 1.000 0.301 7 1.000 0.358 7 1.000 0.707 7 1.000 0.841 7 1.000 0.426 229 110 7 1.000 0.304 7 1.000 0.353 7 1.000 0.708 7 1.000 0.860 7 1.000 0.429 229 120 7 1.000 0.306 7 1.000 0.361 7 1.000 0.710 7 0.999 0.848 7 1.000 0.431 230 50 7 1.000 o. 297 7 1.000 0.360 7 1.000 0.727 7 1.000 0.823 7 1.000 0.408 230 60 7 1.000 0.285 7 1.000 0.366 7 1.000 0.733 7 1.000 D.778 7 1.000 0.389 230 70 7 1.000 0.286 7 1.000 0.368 7 1.000 0.745 7 1.000 0.778 7 1.000 0.384 230 80 7 1.000 0.272 7 1.000 0.376 7 1.000 0.763 7 1.000 0.724 7 1.000 0.357 230 90 7 1.000 0.265 7 1.000 0.384 7 1.000 0. 773 7 1.000 0.691 7 1.000 0.343 230 100 7 1.000 0.260 7 1.000 0.394 7 1.000 0.784 7 1.000 0.659 7 1.000 0.331 231 50 5 0.999 0.236 5 1.000 0.369 5 1.000 0.728 5 0.999 0.639 5 1.000 0.324 231 60 5 0.999 0.253 5 1.000 0.358 5 1.000 0.720 5 1.000 0.706 5 0.999 0.351 231 70 5 0.999 0.249 5 1.000 0.366 5 1.000 0.728 5 0.999 0.678 5 0.999 0.342 231 80 5 0.999 0.247 5 1.000 0.361 5 1.000 0.729 5 0.999 0.685 5 0.999 0.340 231 90 5 1.000 0.253 5 1.000 0.360 5 1.000 0. 726 5 0.999 0.702 5 1.000 0.348 231 100 5 1.000 0.249 5 1.000 0.355 5 1.000 0.724 5 1.000 0.702 5 1.000 0.345 232 40 7 1.000 0.293 7 1.000 0.337 7 1.000 0.661 7 1.000 0.870 7 1.000 0.443 232 50 7 1.000 0.296 7 1.000 0.337 7 1.000 0.666 7 1.000 0.877 7 1.000 0.445 232 60 7 1.000 0.290 7 1.000 0.330 7 1.000 0.666 7 1.000 0.879 7 1.000 0.436 232 70 7 1.000 0.295 7 1.000 0.331 7 1.000 0.664 7 1.000 0.893 7 1.000 0.445 232 80 7 1.000 0.297 7 1.000 0.328 7 1.000 0.661 7 1.000 0.906 7 1.000 0.449 232 90 7 1.000 0.290 7 1.000 0.329 7 1.000 0.655 7 1.000 0.883 7 1.000 0.443 232 100 7 1.000 0.292 7 1.000 0.327 7 1.000 0.656 7 1.000 0.892 7 1.000 0.444 ' 233 50 7 0.999 0.247 7 1.000 0.346 7 1.000 0.701 7 0.999 0.714 7 1.000 0.352 233 60 7 1.000 0.249 7 1.000 0.349 7 1.000 0.697 7 1.000 0. 712 7 1.000 0.356 233 70 7 1.000 0.255 7 1.000 0.351 7 1.000 0.703 7 1.000 0.727 7 1.000 0.363 233 80 7 1.000 0.251 7 1.000 0.350 7 1.000 o. 703 7 1.000 0.718 7 1.000 0.357 233 90 7 1.000 0.247 7 1.000 0.353 7 1.000 0.699 7 1.000 0.698 7 1.000 0.353 233 100 7 1.000 0.252 7 1.000 0.353 7 1.000 0.702 7 1.000 0. 715 7 1.000 0.360 234 70 5 0.999 0.292 5 0.996 0.315 5 0.999 0. 768 5 0.995 0.925 5 0.999 0.380 234 84 7 0.990 0.236 7 1.000 0.377 5 0.999 0.916 7 0.985 0.629 5 0.998 0.288 234 99 7 0.989 0.192 7 0.999 0.403 5 0.996 0.765 7 0.984 0.475 5 0.996 0.277 235 40 18 0.997 0.305 18 0.998 0.366 18 0.999 0.735 18 0.997 0.835 18 0.998 0.416 235 50 18 0.991 0.331 18 0.999 0.382 18 0.998 0.761 18 0.996 0.872 18 0.997 0.437 235 60 18 0.994 0.321 18 0.998 0.382 18 0.999 0.756 18 0.998 0.843 18 0.998 0.426 235 70 18 0.994 0.320 18 0.998 0.379 18 0.999 0. 751 18 0.998 0.847 18 0.998 0.427 235 80 18 0.994 0.310 18 0.999 0.365 18 0.999 0.739 18 0.997 0.853 18 0.997 0.421 235 90 18 0.994 0.309 18 0.998 0.358 18 0.999 0.725 18 0.997 0.865 18 0.997 0.426 235 100 18 0.995 0.308 18 0.999 0.353 18 0.999 0. 721 18 0.997 0.873 18 0.998 0.427 235 110 18 0.994 0.306 18 0.999 0.354 18 0.999 0.723 18 0.997 0.868 18 0.996 0.425 236 40 9 0.994 0.319 9 1.000 0.380 9 1.000 0.749 9 0.993 0.844 9 0.993 0.426 236 50 9 0.995 0.292 9 1.000 0.371 9 1.000 0.745 9 0.993 0.788 9 0.993 0.391 236 60 9 0.997 0.272 9 1.000 0.375 9 1.000 0.747 9 0.997 0. 725 9 0.996 0.364 236 70 9 0.997 0.266 9 1.000 0.379 9 1.000 0.747 9 0.997 0.702 g 0.996 0.355 234 Table 6.3. (continued) Allometric Relationships Fit bX Principal Component Analxsi sc log ii - log w log OBH - log w log BSLA - log w log ii - log DSH log ii - log BSLA !d. " $ow " " 1\ Codea Cond ,b r hw <~>sw r 4>ho n r hB 236 80 10 0.998 0.267 10 1.000 0.381 10 1.000 0.757 10 0.998 0.701 10 0.998 0.352 237 1 12 0.999 0.290 12 1.000 0.373 12 1.000 0. 752 12 0.999 0.778 12 1.000 0.386 237 2 8 1.000 0.300 8 0.999 0.349 8 1.000 0.706 8 0.999 0.859 8 1.000 0.425 237 3 8 1.000 0.322 8 1.000 0.333 8 0.999 0.672 8 1.000 0.965 8 1.000 0.479 237 4 8 1.000 0.340 8 0.999 0.325 8 0.999 0.638 8 0.999 1.046 8 1.000 0.532 237 5 5 1.000 0.348 5 0.999 0.295 5 0.999 0.579 5 0.999 1.179 5 0.999 0.600 238 40 5 0.998 0.257 5 0.999 0.399 5 1.000 0.779 5 0.995 0.642 5 0.998 0.329 238 50 5 0.998 0.258 5 1.000 0.393 5 1.000 0.780 5 0.999 0.658 5 0.997 0.331 238 60 5 1.000 0.266 5 1.000 0.395 5 1.000 0.782 5 1.000 0.672 5 0.999 0.340 238 70 5 0.997 0.258 5 1.000 0.393 5 1.000 o. 781 5 0.996 0.656 5 0.997 0.330 239 40 5 0.993 0.220 5 1.000 0.362 5 1.000 0.736 5 0.990 0.609 5 0.991 0.299 239 50 5 0.999 0.221 5 1.000 0.368 5 1.000 0.739 5 0.999 0.600 5 0.999 0.299 239 60 5 0.998 0.235 5 0.999 0.374 5 1.000 0.738 5 0.996 0.629 5 0.997 0.318 239 70 5 0.996 0.241 5 1.000 0.372 5 1.000 0.737 5 0.995 0.648 5 0.994 0.327 241 20 5 0.986 0,305 5 0.984 0.221 5 0.992 0.330 5 0.960 1.393 5 0.980 0.929 241 30 5 0.999 0.333 5 0.994 0.210 5 0.998 0.332 5 0.993 1.582 5 0.998 1.002 241 40 5 0.993 0.331 5 0.992 0.210 5 0.994 Q.344 5 0.993 1 .573 5 0.976 0.961 241 50 5 0.997 0.327 5 0.997 0.210 5 0.997 0.350 5 0.999 1.560 5 0.989 0.934 241 60 5 0.998 0.358 5 0.994 0.203 5 0.991 0.337 5 0.994 1.762 5 0.983 1.058 242 56 3 1.000 0.285 3 0.980 0.810 3 0.986 1.670 3 0.981 0.345 3 0.987 0.169 242 73 4 0.967 0.317 4 0.995 0.545 4 0.996 1.093 4 0.987 0.593 4 0.985 0.294 242 90 4 0.999 0.258 4 0.999 0.473 4 1.000 0.931 4 0.999 0.546 4 0.999 0.277 ?~ 243 1 7 0.998 0.225 243 2 5 0.988 0.179 244 70 3 0.999 0.328 3 0.998 0.388 3 0.999 0.763 3 1.000 0.845 3 1.000 0.430 244 86 4 0.999 0.329 4 1.000 0.411 4 1.000 0.837 4 0.999 0.801 4 1.000 0.393 244 102 5 0.999 0.315 5 0.999 0.417 5 0.999 0.844 5 1.000 0.754 5 1.000 0.373 245 1 5 0.999 0.388 5 1.000 0.804 245 2 5 1.000 0.454 5 1.000 0.893 245 3 4 0.995 0.490 4 1.000 0.942 246 30 4 0.997 0.296 4 0.999 0.365 4 1.000 0.725 4 0.995 0.812 4 0.995 0.408 246 50 4 0.995 0.217 4 1.000 0.409 4 1.000 0.804 4 0.993 0.532 4 0.992 0.269 246 60 4 0.999 0.176 4 1.000 0.427 4 1.000 0.855 4 0.999 0.412 4 0.999 0.206 246 70 4 0.998 o. 173 4 1.000 0.431 4 1.000 0.855 4 0.998 0.402 4 0.998 0.203 246 80 4 0.998 0.191 4 1.000 0.419 4 1.000 0.841 4 0.998 0.455 4 0.997 0.227 247 1 4 1.000 0.390 4 0.995 0.333 4 1.000 0.689 4 0.994 1.167 4 1.000 0.566 247 2 4 1.000 0.408 4 0.999 0.316 4 1.000 0.635 4 1.000 1.290 4 0.999 0.643 247 3 4 1.000 0.426 4 0.999 0.298 4 1.000 0.607 4 0.999 1.430 4 1.000 0.702 247 4 4 1.000 0.438 4 1.000 0.295 4 1.000 0.584 4 1.000 1.483 4 1.000 0. 750 248 1 5 0.999 0.197 5 0.995 0.248 5 0.998 0.793 248 2 6 0.996 0.191 6 0.999 0.241 6 0.996 0.792 248 3 7 0.996 0.195 7 0.998 0.248 7 0.998 0.78R ?49 100 3 0.995 0.151 3 1.000 0.399 3 1.000 0.875 3 0.996 0.380 3 0.994 0.173 235 Table 8.3. (cant i nued) Allometric Relationships Fit b;t Principal Component Ana l;ts is c log h - log w log DBH - log w log BSLA - log w log h - log DBH log fi - 1 og Bill ld. A A A Code3 Cond.b ~hw ~Ow ~Bw $hD n r $h8 249 110 3 1.000 o. 158 3 0.999 0.409 3 1.000 0.874 3 0.999 0.386 3 1.000 0.181 249 120 3 0.998 0.155 3 1.000 0.365 3 1.000 0.881 3 0.997 0.425 3 0.998 0.176 250 2 4 0.999 0.442 250 3 8 0.999 0.332 251 1 27 1.000 0.315 27 1.000 0.394 27 0.999 0.800 252 120 3 0.999 0.342 3 1.000 0.307 3 1.000 0.598 3 0.997 1.115 3 0.998 0.571 252 140 5 1.000 0.367 5 1.000 0.299 5 0.999 0.601 5 0.999 1.228 5 0.999 0.611 252 160 9 0.999 0.379 9 0.999 0.312 9 0.991 0.704 9 1.000 1.214 9 0.994 0.535 252 180 9 0.999 0.384 9 0.998 0.333 9 0.998 0.665 9 0.999 1.152 9 0.999 0.578 252 200 9 0.999 0.376 9 0.997 0.341 9 0.997 0.683 9 0.999 1.101 9 0.999 0.550 252 220 9 0.999 0.368 9 0.997 0.346 9 0.996 0.687 9 0.999 1.064 9 0.998 0.535 252 240 9 0.999 0.357 9 0.997 0.348 9 0.996 0.698 9 0.998 1.024 9 0.998 0.510 253 47 10 0.999 0.173 10 1.000 0.251 10 1.000 0.582 10 1.000 0.690 10 0.999 0.297 253 57 13 0.999 o. 192 13 1.000 0.259 13 0.999 0.668 13 0.999 0. 740 13 0.999 0.287 '( 253 66 14 0.994 0.192 14 0.998 0.259 14 0.999 0.721 14 0.999 0.742 14 0.996 0.266 254 50 12 1.000 0.212 12 0.997 0.238 12 1.000 0.651 12 0.998 0.888 12 0.999 0.325 254 61 9 0.999 0.247 9 1.000 0.287 9 0.998 0.688 9 0.999 0.862 9 0.997 0.359 254 70 11 0.999 0.275 11 0.998 0.307 11 0.999 0.684 11 0.999 0.896 11 1.000 0.403 255 87 7 0.996 0.283 7 1.000 0.385 7 0.996 0.738 256 40 4 1.000 0.308 4 1.000 0.270 4 1.000 0.544 4 1.000 1.139 4 1.000 0.566 256 53 4 1.000 0.322 4 1.000 0.292 4 1.000 0.575 4 1.000 1.101 4 1.000 0.559 256 66 4 0.999 0.327 4 1.000 0.308 4 0.999 0.617 4 1.000 1.062 4 1.000 0.530 257 45 13 0.986 0.301 13 1.000 0.365 13 1.000 0.726 13 0.989 0.833 13 0.990 0.416 257 49 11 0.997 0.317 11 1.000 0.385 11 1.000 o. 762 11 0.997 0.823 11 0.997 0.416 257 53 11 0.998 0.361 11 0.999 0.344 11 1.000 0.689 11 0.998 1.049 11 0.999 0.524 257 54 11 1.000 0.304 11 1.000 0.357 11 1.000 o. 707 11 0,999 0.853 11 0.999 0.431 257 57 10 0.999 0.517 10 0.999 0.351 10 1.000 0.689 10 0.999 1.474 10 0.999 0. 751 258 40 6 0.999 0.306 6 0.999 0.360 6 0.999 o. 725 6 0.999 0.850 6 0.999 0.422 258 52 6 0.999 0.301 6 1.000 0.333 6 1.000 0.664 6 0.999 0.904 6 0.999 0.453 258 60 6 0.959 0.368 6 1.000 0.336 6 1.000 0.669 6 0.960 1.135 6 0.960 0.555 259 100 14 0.985 0.261 14 1.000 0.402 14 1.000 0.810 14 0.981 0.652 14 0.980 0.321 259 110 13 0.987 0.243 13 0.999 0.412 13 1.000 0.830 13 0.982 0.591 13 0.985 0.293 259 120 13 0.987 0.245 13 0.999 0.414 13 1.000 0.839 13 0.984 0.594 13 0.984 0.292 259 130 13 0.985 0.246 13 1.000 0.415 13 1.000 0.845 13 0.982 0.596 13 0.982 0.291 259 140 13 0.983 0.248 13 1.000 0.423 13 1 .000 0.850 13 0.979 0.589 13 0.980 0.291 259 150 13 0.986 0.246 13 1.000 0.426 13 1.000 0.857 13 0. 983 0.580 13 0.984 0.287 259 160 13 0.984 0.249 13 1.000 0.428 13 1.000 0.858 13 0.981 0.584 13 0.982 0.290 259 170 13 0.985 0.249 13 1.000 0.427 13 1.000 0.856 13 0.983 0.585 13 0.982 0.290 259 180 13 0.983 0.250 13 1.000 0.429 13 1.000 0.861 13 0.980 0.586 13 0.981 0.290 259 190 13 0.984 0.253 13 1.000 0.430 13 1.000 0.863 13 0. 981 0.592 13 0.982 0.293 259 200 13 0.984 0.255 13 1.000 0.434 13 1.000 0.865 13 0.982 0.591 13 0.982 0.294 259 210 13 0.984 0.252 13 1.000 0.433 13 1.000 0.868 13 0.982 0.586 13 0.982 0.290 260 60 7 0.991 0.231 7 1.000 0.386 7 1.000 0. 766 7 0.990 0.601 7 0.990 0.302 236 Table 8.3. (cant i nued) "' Allometric Relationships Fit b;t Principal Component Anal;tsi sc log fi - log w log DBH - log w log BSLA - log w log fi - log DBH log fi - 1 og BSLA !d. A A A A A Codea Cond. b hw <~>ow <~>sw hO hB 260 70 9 0.990 0.249 9 0.999 0.372 9 1 .000 0.773 9 0.987 0.671 9 0.990 0.322 260 80 9 0. 986 0.247 9 0.993 0.413 9 1.000 o. 776 9 0.978 0.595 9 0.984 0.318 260 90 9 0.992 0.238 9 0.999 0.395 9 1 .000 0.790 9 0.988 0.603 q 0.990 0.301 260 100 9 0.989 0.243 9 1.000 0.403 9 1.000 0.810 9 0.986 0.606 9 0.987 0.300 260 110 9 0.989 0.245 9 0.999 0.412 9 1.000 0.820 9 0.984 0.595 9 0.985 0.298 260 120 9 0.986 0.249 9 1.000 0.430 9 1.000 0.844 9 0.987 0. 583 9 0.983 0.295 260 130 9 0.987 0.244 9 1.000 0.425 9 1.000 0.847 9 0.988 0.578 9 0.985 0.288 260 140 9 0.985 0.251 9 0.999 0.441 9 1.000 0.858 9 0.988 0.574 9 0.983 0.293 260 150 9 0.985 0.244 9 1.000 0.423 9 1.000 0.857 9 0.985 0.579 9 0.984 0.284 260 160 9 0.985 0.245 9 1.000 0.425 9 1.000 0.860 9 0.986 0.580 9 0.985 0.285 260 170 9 0.981 0.253 9 0.998 0.440 9 1.000 0.857 9 0.987 0.580 9 0.979 0.295 260 180 9 0.988 0.244 9 1.000 0.427 9 1.000 0.864 9 0.989 0.574 9 0.988 0.282 261 60 8 0.978 0.227 8 1.000 0.366 8 1.000 0.745 8 0.975 0.625 8 0.980 0.306 261 70 8 0.985 0.226 8 0.999 0.362 8 1.000 0.755 8 0.979 /0.625 8 0.986 0.299 261 80 9 0.989 0.246 9 0.993 0.400 9 1.000 0. 750 9 0.980 0.611 9 0.986 0.327 261 90 9 0.989 0.233 9 0.999 0.387 9 1.000 0.774 9 0.985 0.604 9 0.988 0.301 261 100 9 0.990 0.233 9 1.000 0.390 9 1.000 0.784 9 0.986 0.599 9 0.987 0.296 261 110 9 0.992 0.247 9 0.999 0.400 9 1.000 o. 796 9 0.991 0.619 9 0.991 0.311 261 120 9 0.991 0.254 9 1.000 0.412 9 1.000 0.807 9 0.991 0.620 9 0.988 0.314 261 130 9 0. 991 0.247 9 1.000 0.414 9 1.000 0.824 9 0.991 0.599 9 0.988 0.299 261 140 9 0.988 0.253 9 1.000 0.429 9 1.000 0.834 9 0.989 0.594 9 0.984 0.303 261 150 9 0.988 0.246 9 1.000 0.412 9 1.000 0.834 9 0.987 0.600 9 0.986 0.295 261 160 9 0.987 0.243 9 1.000 0.413 g 1.000 0.836 9 0.988 0.592 9 0.988 0.291 262 70 13 0.996 0.361 13 0. 998 0.388 13 1.000 0.740 13 0.990 0.929 13 0.996 0.487 ? 262 80 13 0.996 0.363 13 0.997 0.382 13 1.000 0.742 13 0.985 0.949 13 0.996 0.489 262 90 13 0.996 0.368 13 0.998 0.391 13 1.000 0. 737 13 0.988 0.940 13 0.996 0.499 262 100 13 0.996 0.365 13 0.997 0.383 13 1.000 0.737 13 0.988 0.953 13 0.996 0.495 262 110 13 0.996 0.370 13 0.997 0.386 13 1.000 0. 736 13 0.986 0.960 13 0.995 0.503 262 120 13 0.996 0.368 13 0.996 0.381 13 1.000 o. 735 13 0.985 0.964 13 0.996 0.501 262 130 13 0.996 0.368 13 0.997 0.385 13 1.000 0.735 13 0.986 0.955 13 0.996 0.500 262 140 13 0.996 0.372 13 0.997 0.38.8 13 1.000 0.736 13 0.986 0.958 13 0.996 0.506 262 150 13 o. 991 0.359 13 0.996 0.380 13 0.997 0. 722 13 0.982 0.949 13 0,991 0.498 263 26 4 1.000 0.250 4 0.994 0.275 4 0.996 0.596 4 0.990 0.905 4 0.994 0.418 263 33 7 0.998 0.281 7 0.999 0.261 7 0.999 0.578 7 0.997 1.079 7 0.997 0.487 263 39 9 0.999 0.317 9 1.000 0.269 9 0.999 0.600 9 1 .ooo 1.179 9 0.999 0.527 264 40 4 0.998 0.128 4 0.993 o. 172 4 0.999 0.838 4 0.998 o. 738 4 0.995 0.152 264 50 7 0.994 0.182 7 1.000 0.232 7 1.000 0. 740 7 0.995 0.786 7 0.991 0.245 265 77 4 0.965 0.137 4 1.000 0.419 4 1.000 0.840 4 0.958 0.327 4 0.959 0.163 266 60 8 0.996 0.209 8 1.000 0.438 8 1.000 0.873 8 0.997 0.478 8 0.996 0.240 ?66 80 11 0.997 0.238 11 1.000 0.431 11 1.000 0.862 11 0.995 0. 553 11 0.996 0.276 266 100 11 0.993 0.232 11 1.000 0.415 11 1.000 0.833 11 0.993 0.559 11 0.993 0.278 266 120 12 0.984 0.245 12 1.000 0.410 12 1.000 0.821 12 0.984 0.601 12 0.984 0.298 266 140 8 0.995 0.342 8 1.000 0.385 8 0.999 0.770 8 0.992 0.891 8 0.992 0.443 237 Table 8.3. (continued) Allometric Relationships Fit by Principal Component Analysi sc log ii - log w log DBH - log w log BSLA - log w log ii - log DBH log ii - log BSLA !d. " " " " " Codea Cond .b r hw n r <~>ow r sw r hO r hB 277 120 7 0.997 0.195 7 0.999 0.349 7 0.999 0.760 7 1.000 0.560 7 0.994 0.256 278 1 19 0.999 0.237 19 1.000 0.435 19 1.000 0.870 19 0.999 0.547 19 0.999 0.273 279 40 10 0.996 0.202 10 1.000 0.392 10 1.000 0.788 10 0.994 0.515 10 0.995 0.256 280 51 6 0.998 0.238 6 0.999 0.431 6 0.999 0.862 6 0.999 0.552 6 0.999 0.276 281 27 5 1.000 0.393 5 1.000 0.786 281 30 5 1.000 0.401 5 1.000 0.801 281 33 5 1.000 0.402 5 1.000 0.807 aTable B.l associates each !d. code with a particular yield table. bsee Table B.l and the references given there for further information on condition ? ...S:he general..f.Qrmula for the allometric relation is log_y_ = $1 ].Qg_X + $o, where Y is l,og fi, log DBH, or log BSLA and X is log w, log OBH, or log BSCA. OBH and BSLA are, respectively, the diameter at breast height and the basal area at breast height of the boles of individual trees. Y and X are paired as indicated in the table heading. The number of data points, correlation (r), and slope for each relationship are given. The correlation values have little statistical meaning because most of the natural variability in the original forest data has been removed from the yield table predictions. They ar~ inc 1 uded here only as a crude index of variation around the fitted 1 i ne. Va 1 ues of the intercept,