LETTER
doi:10.1038/nature12914
Rate of tree carbon accumulation increases
continuously with tree size
N. L. Stephenson1, A. J. Das1, R. Condit2, S. E. Russo3, P. J. Baker4, N. G. Beckman3{, D. A. Coomes5, E. R. Lines6, W. K. Morris7,
N. Ru¨ger2,8{, E. A´lvarez9, C. Blundo10, S. Bunyavejchewin11, G. Chuyong12, S. J. Davies13, A´. Duque14, C. N. Ewango15, O. Flores16,
J. F. Franklin17, H. R. Grau10, Z. Hao18, M. E. Harmon19, S. P. Hubbell2,20, D. Kenfack13, Y. Lin21, J.-R. Makana15, A. Malizia10,
L. R.Malizia22, R. J. Pabst19, N. Pongpattananurak23, S.-H. Su24, I-F. Sun25, S. Tan26, D. Thomas27, P. J. vanMantgem28, X.Wang18,
S. K. Wiser29 & M. A. Zavala30
Forests aremajor components of the global carbon cycle, providing
substantial feedback to atmospheric greenhouse gas concentrations1.
Our ability to understand and predict changes in the forest carbon
cycle—particularly net primary productivity and carbon storage—
increasingly relies on models that represent biological processes
across several scales of biological organization, from tree leaves to
forest stands2,3. Yet, despite advances in our understanding of pro-
ductivity at the scales of leaves and stands, no consensus exists about
the nature of productivity at the scale of the individual tree4–7, in
part because we lack a broad empirical assessment of whether rates
of absolute treemassgrowth(and thuscarbonaccumulation)decrease,
remain constant, or increase as trees increase in size and age. Here we
present a global analysis of 403 tropical and temperate tree species,
showing that for most species mass growth rate increases continu-
ously with tree size. Thus, large, old trees do not act simply as se-
nescent carbon reservoirs but actively fix large amounts of carbon
compared to smaller trees; at the extreme, a single big tree can add
the same amount of carbon to the forest within a year as is contained
in an entire mid-sized tree. The apparent paradoxes of individual
tree growth increasing with tree size despite declining leaf-level8–10
and stand-level10 productivity can be explained, respectively, by
increases in a tree’s total leaf area that outpace declines in produc-
tivity per unit of leaf area and, among other factors, age-related
reductions in population density. Our results resolve conflicting
assumptions about thenatureof tree growth, informefforts tounder-
tand andmodel forest carbon dynamics, and have additional impli-
cations for theories of resource allocation11 and plant senescence12.
Awidely held assumption is that after an initial period of increasing
growth, the mass growth rate of individual trees declines with increas-
ing tree size4,5,13–16. Although the results of a few single-species studies
have been consistent with this assumption15, the bulk of evidence cited
in support of declining growth is not based on measurements of indi-
vidual treemass growth. Instead,muchof the cited evidencedocuments
either the well-known age-related decline in net primary productivity
(hereafter ‘productivity’) of even-aged forest stands10 (in which the trees
are all of a similar age) or size-related declines in the rate ofmass gain per
unit leaf area (or unit leaf mass)8–10, with the implicit assumption that
declines at these scales must also apply at the scale of the individual tree.
Declining tree growth is also sometimes inferred from life-history theory
to be a necessary corollary of increasing resource allocation to reproduc-
tion11,16. On the other hand, metabolic scaling theory predicts that mass
growth rate should increase continuously with tree size6, and this pre-
diction has also received empirical support from a few site-specific
studies6,7. Thus, we are confronted with two conflicting generalizations
about the fundamental nature of tree growth, but lack a global assess-
ment that would allow us to distinguish clearly between them.
To fill this gap, we conducted a global analysis in which we directly
estimated mass growth rates from repeated measurements of 673,046
trees belonging to 403 tropical, subtropical and temperate tree species,
spanning every forested continent. Tree growth rate wasmodelled as a
function of log(tree mass) using piecewise regression, where the inde-
pendent variable was divided into one to four bins. Conjoined line
segments were fitted across the bins (Fig. 1).
For all continents, aboveground tree mass growth rates (and, hence,
rates of carbon gain) for most species increased continuously with tree
mass (size) (Fig. 2). The rate of mass gain increased with tree mass in
eachmodel bin for 87%of species, and increased in the bin that included
the largest trees for 97% of species; the majority of increases were sta-
tistically significant (Table 1, ExtendedData Fig. 1 and Supplementary
Table 1). Even whenwe restricted our analysis to species achieving the
largest sizes (maximum trunkdiameter.100 cm; 33%of species), 94%
had increasing mass growth rates in the bin that included the largest
trees.We found no clear taxonomic or geographic patterns among the
3%of specieswith declining growth rates in their largest trees, although
the small number of these species (thirteen)hampers inference.Declin-
ing species included both angiosperms and gymnosperms in seven of
the 76 families in our study; most of the seven families had only one or
two declining species and no family was dominated by declining spe-
cies (Supplementary Table 1).
Whenwe log-transformedmass growth rate in addition to treemass,
the resulting model fits were generally linear, as predicted bymetabolic
scaling theory6 (ExtendedData Fig. 2). Similar to the results of ourmain
1USGeological Survey,Western Ecological ResearchCenter, Three Rivers, California 93271, USA. 2Smithsonian Tropical Research Institute, Apartado 0843-03092, Balboa, Republic of Panama. 3School of
Biological Sciences, University ofNebraska, Lincoln,Nebraska68588,USA. 4Department of Forest andEcosystemScience, University ofMelbourne, Victoria 3121, Australia. 5Department of Plant Sciences,
University of Cambridge, Cambridge CB2 3EA, UK. 6Department of Geography, University College London, London WC1E 6BT, UK. 7School of Botany, University of Melbourne, Victoria 3010, Australia.
8Spezielle Botanik und Funktionelle Biodiversita¨t, Universita¨t Leipzig, 04103 Leipzig, Germany. 9Jardı´n Bota´nico de Medellı´n, Calle 73, No. 51D-14, Medellı´n, Colombia. 10Instituto de Ecologı´a Regional,
UniversidadNacional de Tucuma´n, 4107 Yerba Buena, Tucuma´n, Argentina. 11ResearchOffice, Department of National Parks, Wildlife and Plant Conservation, Bangkok 10900, Thailand. 12Department of
Botany and Plant Physiology, Buea, Southwest Province, Cameroon. 13Smithsonian Institution Global Earth Observatory—Center for Tropical Forest Science, Smithsonian Institution, PO Box 37012,
Washington, DC 20013, USA. 14Universidad Nacional de Colombia, Departamento de Ciencias Forestales, Medellı´n, Colombia. 15Wildlife Conservation Society, Kinshasa/Gombe, Democratic Republic of
the Congo. 16Unite´ Mixte de Recherche—Peuplements Ve´ge´taux et Bioagresseurs en Milieu Tropical, Universite´ de la Re´union/CIRAD, 97410 Saint Pierre, France. 17School of Environmental and Forest
Sciences, University of Washington, Seattle, Washington 98195, USA. 18State Key Laboratory of Forest and Soil Ecology, Institute of Applied Ecology, Chinese Academy of Sciences, Shenyang 110164,
China. 19Department of Forest Ecosystems and Society, Oregon State University, Corvallis, Oregon 97331, USA. 20Department of Ecology and Evolutionary Biology, University of California, Los Angeles,
California 90095, USA. 21Department of Life Science, Tunghai University, Taichung City 40704, Taiwan. 22Facultad de Ciencias Agrarias, Universidad Nacional de Jujuy, 4600 San Salvador de Jujuy,
Argentina. 23Faculty of Forestry, Kasetsart University, ChatuChak Bangkok 10900, Thailand. 24Taiwan Forestry Research Institute, Taipei 10066, Taiwan. 25Department of Natural Resources and
Environmental Studies, National Dong Hwa University, Hualien 97401, Taiwan. 26Sarawak Forestry Department, Kuching, Sarawak 93660, Malaysia. 27Department of Botany and Plant Pathology, Oregon
State University, Corvallis, Oregon 97331, USA. 28US Geological Survey, Western Ecological Research Center, Arcata, California 95521, USA. 29Landcare Research, PO Box 40, Lincoln 7640, New Zealand.
30Forest Ecology and Restoration Group, Department of Life Sciences, University of Alcala´, Alcala´ de Henares, 28805 Madrid, Spain. {Present addresses: Mathematical Biosciences Institute, Ohio State
University, Columbus, Ohio 43210, USA (N.G.B.); German Centre for Integrative Biodiversity Research (iDiv), Halle-Jena-Leipzig, 04103 Leipzig, Germany (N.R.).
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analysis using untransformed growth, of the 381 log-transformed spe-
cies analysed (seeMethods), the log-transformed growth rate increased
in the bin containing the largest trees for 96% of species.
In absolute terms, trees 100 cm in trunk diameter typically add from
10kg to200kgof abovegrounddrymass eachyear (dependingonspecies),
averaging 103 kg per year. This is nearly three times the rate for trees of
the same species at 50 cm in diameter, and is the mass equivalent to
adding an entirely new tree of 10–20 cm in diameter to the forest each
year.Our findings further indicate that the extraordinary growth recently
reported in an intensive study of large Eucalyptus regnans and Sequoia
sempervirens7, which included some of the world’s most massive indi-
vidual trees, is not a phenomenon limited to a fewunusual species. Rather,
rapid growth in giant trees is the global norm, and can exceed 600 kg
per year in the largest individuals (Fig. 3).
Our data set includedmanynatural andunmanaged forests inwhich
the growth of smaller trees was probably reduced by asymmetric com-
petition with larger trees. To explore the effects of competition, we cal-
culatedmass growth rates for 41NorthAmerican andEuropean species
that had published equations for diameter growth rate in the absence of
competition. We found that, even in the absence of competition, 85%
of the species hadmass growth rates that increased continuouslywith tree
size (Extended Data Fig. 3), with growth curves closely resembling those
in Fig. 2. Thus, our finding of increasing growth not only has broad
generalityacross species, continentsandforestbiomes (tropical, subtropical
and temperate), it appears tohold regardless of competitive environment.
Importantly, our finding of continuously increasing growth is com-
patiblewith the two classes of observationsmost often cited as evidence
of declining, rather than increasing, individual tree growth:with increas-
ing tree size and age, productivity usually declines at the scales of both
tree organs (leaves) and tree populations (even-aged forest stands).
First, although growth efficiency (treemass growth per unit leaf area
or leaf mass) often declines with increasing tree size8–10, empirical
observations and metabolic scaling theory both indicate that, on aver-
age, total tree leaf mass increases as the square of trunk diameter17,18. A
typical tree that experiences a tenfold increase in diameter will therefore
undergo a roughly100-fold increase in total leafmass and a 50–100-fold
–1
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Trunk diameter (cm)
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20 40
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10 10020 40
Figure 1 | Example model fits for tree mass growth rates. The species shown
are the angiosperm species (Lecomtedoxa klaineana, Cameroon, 142 trees) (a)
and gymnosperm species (Picea sitchensis, USA, 409 trees) (b) in our data
set that had the most massive trees (defined as those with the greatest
cumulative aboveground dry mass in their five most massive trees). Each point
represents a single tree; the solid red lines represent best fits selected by our
model; and the dashed red lines indicate one standard deviation around the
predicted values.
10 100 10 100 100
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log10 [mass (Mg)]
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a b c20 40 20 40 10 20 40
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10 100 10 100 100d e f20 40 20 40 10 20 40
–1 2 –2 0 1–1 2 –2 0 1–1 2
Figure 2 | Aboveground mass growth rates for the 403 tree species, by
continent. a, Africa (Cameroon, Democratic Republic of the Congo); b, Asia
(China, Malaysia, Taiwan, Thailand); c, Australasia (New Zealand); d, Central
and South America (Argentina, Colombia, Panama); e, Europe (Spain); and
f, NorthAmerica (USA).Numbers of trees, numbers of species and percentages
with increasing growth are given in Table 1. Trunk diameters are approximate
values for reference, based on the average diameters of trees of a given mass.
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increase in total leaf area (depending on size-related increases in leaf
mass per unit leaf area19,20). Parallel changes in growth efficiency can
range from a modest increase (such as in stands where small trees are
suppressed by large trees)21 to asmuch as a tenfold decline22, withmost
changes falling in between8,9,19,22. At one extreme, the net effect of a low
(50-fold) increase in leaf area combinedwith a large (tenfold) decline in
growth efficiency would still yield a fivefold increase in individual tree
mass growth rate; the opposite extreme would yield roughly a 100-fold
increase. Our calculated 52-fold greater average mass growth rate of
trees 100 cm in diameter compared to those 10 cm in diameter falls
within this range. Thus, although growth efficiency often declines with
increasing tree size, increases in a tree’s total leaf area are sufficient to
overcome this decline and cause whole-tree carbon accumulation rate
to increase.
Second, our findings are similarly compatible with the well-known
age-related decline in productivity at the scale of even-aged forest stands.
Although a reviewofmechanisms is beyond the scope of this paper10,23,
several factors (including the interplay of changing growth efficiency
and tree dominance hierarchies24) can contribute to declining produc-
tivity at the stand scale.We highlight the fact that increasing individual
tree growth rate does not automatically result in increasing stand pro-
ductivity because tree mortality can drive orders-of-magnitude reduc-
tions in population density25,26. That is, even though the large trees in
older, even-aged stands may be growing more rapidly, such stands
have fewer trees. Tree population dynamics, especially mortality, can
thus be a significant contributor to decliningproductivity at the scale of
the forest stand23.
For a large majority of species, our findings support metabolic scal-
ing theory’s qualitative prediction of continuously increasing growth
at the scale of individual trees6, with several implications. For example,
life-history theory often assumes that tradeoffs between plant growth
and reproduction are substantial11. Contrary to some expectations11,16,
our results indicate that for most tree species size-related changes in
reproductive allocation are insufficient to drive long-term declines in
growth rates6. Additionally, declining growth is sometimes considered
to be a defining feature of plant senescence12. Our findings are thus rele-
vant to understanding the nature and prevalence of senescence in the
life history of perennial plants27.
Finally, our results are relevant to understanding and predicting
forest feedbacks to the terrestrial carboncycle andglobal climate system1–3.
These feedbacks will be influenced by the effects of climatic, land-use
and other environmental changes on the size-specific growth rates and
size structure of tree populations—effects that are already being observed
in forests28,29. The rapid growth of large trees indicates that, relative to
their numbers, they could play a disproportionately important role in
these feedbacks30. For example, in our western USA old-growth forest
plots, trees .100 cm in diameter comprised 6% of trees, yet contrib-
uted 33% of the annual forest mass growth. Mechanistic models of the
forest carbon cycle will depend on accurate representation of produc-
tivity across several scales of biological organization, including calibra-
tion and validation against continuously increasing carbon accumulation
rates at the scale of individual trees.
METHODS SUMMARY
Weestimatedabovegrounddrymass growth rates fromconsecutive diametermea-
surements of tree trunks—typically measured every five to ten years—from long-
term monitoring plots. Analyses were restricted to trees with trunk diameter
$10 cm, and to species having$40 trees in total and$15 treeswith trunk diameter
$30 cm.Maximum trunk diameters ranged from 38 cm to 270 cm among species,
averaging 92 cm.We converted each diameter measurement (plus an accompany-
ing height measurement for 16% of species) to aboveground dry mass, M, using
published allometric equations. We estimated tree growth rate asG5DM/Dt and
modelledG as a functionof log(M) for each species using piecewise regression.The
independent variable log(M) was divided into bins and a separate line segmentwas
fitted to G versus log(M) in each bin so that the line segments met at the bin divi-
sions. Bin divisions were not assigned a priori, but were fitted by the model sepa-
rately for each species. We fitted models with 1, 2, 3 and 4 bins, and selected the
model receiving the most support by Akaike’s Information Criterion for each
species. Our approach thus makes no assumptions about the shape of the rela-
tionship between G and log(M), and can accommodate increasing, decreasing or
hump-shaped relationships. Parameters were fitted with a Gibbs sampler based on
Metropolis updates, producingcredible intervals formodel parameters andgrowth
rates at any diameter; uninformative priorswere used for all parameters.We tested
extensively for bias, and found no evidence that our results were influenced by
model fits failing to detect a final growth decline in the largest trees, possible biases
introducedby the 47%of species forwhichwe combined data from several plots, or
possible biases introduced by allometric equations (Extended Data Figs 4 and 5).
Online Content Any additional Methods, ExtendedData display items and Source
Data are available in the online version of the paper; references unique to these
sections appear only in the online paper.
Received 5 August; accepted 27 November 2013.
Published online 15 January 2014.
1. Pan, Y. et al. A large and persistent carbon sink in the world’s forests. Science 333,
988–993 (2011).
Table 1 | Sample sizes and tree growth trends by continent
Continent Number of trees Number of species Percentage of species with increasing mass growth rate in the largest trees
(percentage significant at P#0.05)
Africa 15,366 37 100.0 (86.5)
Asia 43,690 136 96.3 (89.0)
Australasia 45,418 22 95.5 (95.5)
Central and South America 18,530 77 97.4 (92.2)
Europe 439,889 42 90.5 (78.6)
North America 110,153 89 98.9 (94.4)
Total 673,046 403 96.8 (89.8)
The largest trees are those in the last bin fitted by the model. Countries are listed in the legend for Fig. 2.
log10 [mass (Mg)]
–2
M
as
s 
gr
ow
th
 ra
te
 (M
g 
yr
–1
)
0.0
0.2
0.4
0.6
0.8
–1 0 1 2
Figure 3 | Aboveground mass growth rates of species in our data set
compared with E. regnans and S. sempervirens. For clarity, only the 58
species in our data set having at least one tree exceeding 20Mg are shown
(lines). Data for E. regnans (green dots, 15 trees) and S. sempervirens (red dots,
21 trees) are from an intensive study that included some of the most massive
individual trees on Earth7. Both axes are expanded relative to those of Fig. 2.
LETTER RESEARCH
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Supplementary Information is available in the online version of the paper.
Acknowledgements We thank the hundreds of people who have established and
maintained the forest plots and their associated databases; M. G. Ryan for comments
on the manuscript; C. D. Canham and T. Hart for supplying data; C. D. Canham for
discussions and feedback; J. S. Baron for hosting our workshops; and Spain’s
Ministerio de Agricultura, Alimentacio´n y Medio Ambiente (MAGRAMA) for granting
access to the Spanish Forest Inventory Data. Our analyses were supported by the
United States Geological Survey (USGS) John Wesley Powell Center for Analysis and
Synthesis, theUSGSEcosystemsandClimate and LandUseChangemission areas, the
Smithsonian Institution Global Earth Observatory—Center for Tropical Forest Science
(CTFS), and a University of Nebraska-Lincoln Program of Excellence in Population
BiologyPostdoctoral Fellowship (toN.G.B.). In addition, X.W.was supportedbyNational
Natural Science Foundation of China (31370444) and State Key Laboratory of Forest
and Soil Ecology (LFSE2013-11). Data collection was funded by a broad range of
organizations including the USGS, the CTFS, the US National Science Foundation, the
Andrews LTER (NSF-LTERDEB-0823380), the USNational Park Service, the US Forest
Service (USFS), the USFS Forest Inventory and Analysis Program, the John D. and
Catherine T.MacArthur Foundation, the AndrewW.Mellon Foundation,MAGRAMA, the
Council of Agriculture of Taiwan, the National Science Council of Taiwan, the National
Natural Science Foundation of China, the Knowledge Innovation Program of the
Chinese Academy of Sciences, Landcare Research and the National Vegetation Survey
Database (NVS) of New Zealand, the French Fund for the Global Environment and
Fundacio´n ProYungas. This paper is a contribution from the Western Mountain
Initiative, a USGS global change research project. Any use of trade names is for
descriptive purposes only and does not imply endorsement by the USA government.
Author Contributions N.L.S. and A.J.D. conceived the study with feedback from R.C.
and D.A.C., N.L.S., A.J.D., R.C. and S.E.R. wrote the manuscript. R.C. devised the main
analytical approach and wrote the computer code. N.L.S., A.J.D., R.C., S.E.R., P.J.B.,
N.G.B., D.A.C., E.R.L., W.K.M. and N.R. performed analyses. N.L.S., A.J.D., R.C., S.E.R.,
P.J.B., D.A.C., E.R.L., W.K.M., E.A´., C.B., S.B., G.C., S.J.D., A´.D., C.N.E., O.F., J.F.F., H.R.G., Z.H.,
M.E.H., S.P.H., D.K., Y.L., J.-R.M., A.M., L.R.M., R.J.P., N.P., S.-H.S., I-F.S., S.T., D.T., P.J.v.M.,
X.W., S.K.W. andM.A.Z. supplied data and sources of allometric equations appropriate
to their data.
Author Information Fittedmodel parameters for each species have been deposited in
USGS’s ScienceBase at http://dx.doi.org/10.5066/F7JS9NFM. Reprints and
permissions information is available at www.nature.com/reprints. The authors declare
no competing financial interests. Readers are welcome to comment on the online
version of the paper. Correspondence and requests for materials should be addressed
to N.L.S. (nstephenson@usgs.gov).
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METHODS
Data. We required that forest monitoring plots provided unbiased samples of all
living trees within the plot boundaries, and that the trees had undergone two trunk
diameter measurements separated by at least one year. Some plots sampled min-
imally disturbed old (all-aged) forest, whereas others, particularly those associated
with national inventories, sampled forest stands regardless of past management
history. Plots are described in the references cited in Supplementary Table 1.
Our raw data were consecutive measurements of trunk diameter, D, with most
measurements taken 5 to 10 years apart (range, 1–29 years). D was measured at a
standard height on the trunk (usually 1.3–1.4m above ground level), consistent
across measurements for a tree. Allometric equations for 16% of species required, in
addition to consecutivemeasurements ofD, consecutivemeasurements of treeheight.
We excluded trees exhibiting extreme diameter growth, defined as trunkswhere
D increased by $40mmyr21 or that shrank by $12s, where s is the standard
deviation of the D measurement error, s5 0.90361 0.006214D (refs 31, 32); out-
liers of these magnitudes were almost certainly due to error. By being so liberal in
allowing negative growth anomalies, we erred on the side of reducing our ability
to detect increases in tree mass growth rate. Using other exclusion values yielded
similar results, as did a second approach to handling error in which we reanalysed
a subset of our models using a Bayesian method that estimates growth rates after
accounting for error, based on independent plot-specific data quantifying mea-
surement error33.
To standardizeminimumDamongdata sets,weanalysedonly treeswithD$10 cm
at the first census. To ensure adequate samples of trees spanning a broad range of
sizes, we restricted analyses to species having both$40 trees in total and also$15
treeswithD$ 30 cm at the first census. This left us with 673,046 trees belonging to
403 tropical and temperate species in 76 families, spanning twelve countries and all
forested continents (Supplementary Table 1). Maximum trunk diameters ranged
from 38 cm to 270 cm among species, and averaged 92 cm.
Estimating tree mass. To estimate each tree’s aboveground drymass, M, we used
published allometric equations relatingM toD (or for 16%of species, relatingM to
D and tree height). Some equations were species-specific and others were specific
tohigher taxonomic levels or forest types, described in the references in Supplemen-
taryTable 1. The single tropicalmoist forest equation of ref. 34was applied tomost
tropical species, whereasmost temperate species had unique species-specific equa-
tions. Most allometric equations are broadly similar, relating log(M) to log(D)
linearly, or nearly linearly—a familiar relationship in allometric scaling of both
animals and plants35. Equations can show a variety of differences in detail, how-
ever, with some adding log(D) squared and cubed terms. All equationsmake use of
thewooddensity of individual species, butwhenwooddensitywasnot available for
a given species we used mean wood density for a genus or family36.
Using a single, average allometry formost tropical species, andmeanwood den-
sity for a genus or family for several species, limits the accuracy of our estimates of
M. However, becausewe treat each species separately, itmakesnodifferencewhether
our absolute M estimates are more accurate in some species than in others, only
that they are consistent within a species and therefore accurately reveal whether
mass growth rates increase or decrease with tree size.
For two regions—Spain and the western USA—allometric equations estimated
mass only for a tree’s main stem rather than all aboveground parts, including
branches and leaves. But because leaf and stem masses are positively correlated
and their growth rates are expected to scale isometrically both within and among
species18,37,38, results from these two regions should not alter our qualitative con-
clusions. Confirming this, the percentage of species with increasing stem mass
growth rate in the last bin for Spain and the westernUSA (93.4%of 61 species) was
similar to that from the remainder of regions (97.4% of 342 species) (P5 0.12,
Fisher’s exact test).
Modelling mass growth rate. We sought a modelling approach that made no
assumptions about the shape of the relationship between aboveground dry mass
growth rate, G, and aboveground dry mass, M, and that could accommodate
monotonically increasing, monotonically decreasing, or hump-shaped relation-
ships.We therefore chose tomodelG as a functionof log(M) using piecewise linear
regression. The range of the x axis,X5 log(M), is divided into a series of bins, and
within each bin G is fitted as a function of X by linear regression. The position of
the bins is adaptive: it is fitted along with the regression terms. Regression lines are
required to meet at the boundary between bins. For a single model-fitting run the
number of bins, B, is fixed. For example, if B5 2, there are four parameters to be
fitted for a single species: the location of the boundary between bins, X1; the slope
of the regression in the first bin, S1; the slope in the second bin, S2; and an intercept
term. Those four parameters completely define themodel. In general, there are 2B
parameters for B bins.
Growth rates, while approximately normally distributed, were heteroskedastic,
with the variance increasing withmass (Fig. 1), so an additionalmodel was needed
for the standard deviation of G, sG, as a function of log(M). The increase of sG
with log(M) was clearly not linear, so we used a three-parameter model:
sG~k for log Mð Þvdð Þ
sG~azblog Mð Þ (for log Mð Þ§d)
where the intercept a is determined by the values of k, d and b. Thus sG was
constant for smaller values of log(M) (below the cutoff d), then increased linearly
for larger log(M) (Fig. 1). The parameters k, d and bwere estimated along with the
parameters of the growth model.
Parameters of both the growth and standard deviationmodelswere estimated in
a Bayesian framework using the likelihood of observing growth rates given model
predictions and the estimated standard deviation of theGaussian error function. A
Markov chainMonteCarlo chainof parameter estimateswas created using aGibbs
sampler with a Metropolis update39,40 written in the programming language R
(ref. 41) (a tutorial and the computer code are available through http://ctfs.arnarb.
harvard.edu/Public/CTFSRPackage/files/tutorials/growthfitAnalysis). The sampler
works by updating each of the parameters in sequence, holding other parameters
fixed while the relevant likelihood function is used to locate the target parameter’s
next value. The step size used in the updates was adjusted adaptively through the
runs, allowing more rapid convergence40. The final Markov chain Monte Carlo
chain describes the posterior distribution for eachmodel parameter, the error, and
was then used to estimate the posterior distribution of growth rates as estimated
from the model. Priors on model parameters were uniform over an unlimited
range, whereas the parameters describing the standard deviation were restricted
to.0. Bin boundaries, Xi, were constrained as follows: (1) boundaries could only
fall within the range of X, (2) each bin contained at least five trees, and (3) no bin
spanned less than 10% of the range of X. The last two restrictions prevented the
bins fromcollapsing to verynarrow ranges ofX inwhich the fitted slopemight take
absurd extremes.
We chose piecewise regression over other alternatives for modelling G as a
function of M for twomain reasons. First, the linear regression slopes within each
bin provide precise statistical tests of whether G increases or decreases with X,
based on credible intervals of the slope parameters. Second, with adaptive bin
positions, the function is completely flexible in allowing changes in slope at any
point in theX range, with no influence of any one bin on the others. In contrast, in
parametricmodels where a single function defines the relationship across allX, the
shape of the curve at low X can (and indeed must) influence the shape at high X,
hindering statistical inference about changes in tree growth at large size.
We used log(M) as our predictor because within a species M has a highly non-
Gaussian distribution, withmany small trees and only a few very large trees, includ-
ing some large outliers. In contrast,we did not log-transformour dependent variable
G so that we could retain values of G# 0 that are often recorded in very slowly
growing trees, forwhichdiameter change over a shortmeasurement interval can be
on a par with diameter measurement error.
For each species,models with 1, 2, 3 and 4 binswere fitted.Of these fourmodels,
themodel receiving the greatest weight of evidence byAkaike InformationCriterion
(AIC) was selected. AIC is defined as the log-likelihood of the best-fitting model,
penalized by twice the number of parameters. Given that adding onemore bin to a
modelmeant twomore parameters, themodelwith an extra binhad to improve the
log-likelihood by 4 to be considered a better model42.
Assessing model fits. To determine whether our approach might have failed to
reveal a final growth decline within the few largest trees of the various species, we
calculated mass growth rate residuals for the single most massive individual tree
of each species. For 52% of the 403 species, growth of the most massive tree was
underestimated by ourmodel fits (for example, Fig. 1a); for 48% it was overestimated
(for example, Fig. 1b). These proportionswere indistinguishable from50% (P50.55,
binomial test), as would be expected for unbiased model fits. Furthermore, the
mean residual (observedminus predicted)mass growth rate of thesemost massive
trees,10.006Mgyr21, was statistically indistinguishable fromzero (P5 0.29, two-
tailed t-test). We conclude that our model fits accurately represent growth trends
up through, and including, the most massive trees.
Effects of combined data.To achieve sample sizes adequate for analysis, for some
species we combined data from several different forest plots, potentially intro-
ducing a source of bias: if the largest trees of a species disproportionately occur on
productive sites, the increase in mass growth rate with tree size could be exagger-
ated. Thismight occur because trees on less-productive sites—presumably the sites
having the slowest-growing trees within any given size class—could be under-
represented in the largest size classes. We assessed this possibility in two ways.
First, our conclusions remained unchanged when we compared results for the
53% of species that came uniquely from single large plots with those of the 47% of
specieswhose data were combined across several plots. Proportions of species with
increasingmass growth rates in the last binwere indistinguishable between the two
groups (97.6% and 95.8%, respectively; P5 0.40, Fisher’s exact test). Additionally,
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the shapes and magnitudes of the growth curves for Africa and Asia, where data
for each species came uniquely from single large plots, were similar to those of
Australasia, EuropeandNorthAmerica,wheredata for each specieswere combined
across several plots (Table 1, Fig. 2 and Extended Data Fig. 2). (Data from Central
and South America were from both single and combined plots, depending on
species.)
Second, for a subset of combined-data species we compared two sets of model
fits: (1) using all available plots (that is, the analyses we present in the main text),
and (2) using only plots that containedmassive trees—those in the top 5% ofmass
for a species. Tomaximize our ability to detect differences,we limited these analyses
to species with large numbers of trees found in a large number of plots, dispersed
widely across a broad geographic region.We therefore analysed the twelve Spanish
species that each had more than 10,000 individual trees (Supplementary Table 1),
found in 34,580 plots distributed across Spain. Massive trees occurred in 6,588
(19%) of the 34,580 plots. We found no substantial differences between the two
analyses. When all 34,580 plots were analysed, ten of the twelve species showed
increasing growth in the last bin, and seven showed increasing growth across all
bins; when only the 6,588 plots containing the most massive trees were analysed,
the corresponding numbers were eleven and nine. Model fits for the two groups
were nearly indistinguishable in shape andmagnitude across the range of treemasses.
We thus found no evidence that the potential for growth differences among plots
influenced our conclusions.
Effects of possible allometric biases. For some species, the maximum trunk dia-
meterD in our data sets exceeded themaximum used to calibrate the species’ allo-
metric equation. In such cases our estimates of M extrapolate beyond the fitted
allometry and could therefore be subject to bias. For 336 of our 403 species wewere
able to determineDof the largest tree that had beenused in calibrating the associated
allometric equations. Of those 336 species, 74% (dominated by tropical species)
had no trees in our data set withD exceeding that used in calibrating the allometric
equations, with the remaining 26% (dominated by temperate species) having at
least one tree with D exceeding that used in calibration. The percentage of species
with increasing G in the last bin for the first group (98.0%) was indistinguishable
from that of the second group (96.6%) (P5 0.44, Fisher’s exact test). Thus, our
finding of increasingG with tree size is not affected by the minority of species that
have at least one tree exceeding the maximum value of D used to calibrate their
associated allometric equations.
A bias that could inflate the rate at whichG increases with tree size could arise if
allometric equations systematically underestimateM for small trees or overestimate
M for large trees43. For a subset of our study species we obtained the raw data—
consisting of measured values ofD and M for individual trees—needed to calibrate
allometric equations, allowing us to determine whether the particular formof those
species’ allometric equationswasprone tobias, and if so, the potential consequences
of that bias.
To assess the potential for allometric bias for the majority (58%) of species
in our data set—those that used the empirical moist tropical forest equation of
ref. 34—we reanalysed the data provided by ref. 34. The data were from 1,504
harvested trees representing 60 families and 184 genera, withD ranging from 5 cm
to 156 cm; the associated allometric equation relates log(M) to a third-order poly-
nomial of log(D). Because the regression of M on D was fitted on a log–log scale,
this and subsequent equations include a correction of exp[(RSE)2/2] for the error
in back-transformation, where RSE is the residual standard error from the statist-
ical model44. Residuals ofM for the equation revealed no evident biases (Extended
Data Fig. 4a), suggesting that we should expect little (if any) systematic size-related
biases in our estimates of G for the 58% of our species that used this equation.
Our simplest form of allometric equation—applied to 22% of our species—was
log(M)5 a1 blog(D), where a and b are taxon-specific constants. For nine of our
species that used equations of this form (all from the temperate western USA:
Abies amabilis, A. concolor, A. procera, Pinus lambertiana, Pinus ponderosa, Picea
sitchensis, Pseudotsuga menziesii, Tsuga heterophylla and T.mertensiana) we had
values of both D and M for a total of 1,358 individual trees, allowing us to fit
species-specific allometric equations of the form log(M)5 a1 blog(D) and then
assess them for bias. Residual plots showed a tendency to overestimate M for the
largest trees (Extended Data Fig. 4b), with the possible consequence of inflating
estimates of G for the largest relative to the smallest trees of these species.
To determinewhether this bias was likely to alter our qualitative conclusion that
G increases with tree size, we created a new set of allometric relations between D
andM —one for each of the nine species—using the same piecewise linear regres-
sion approach we used to model G as a function of M. However, because our goal
was to eliminate bias rather than seek the most parsimonious model, we fixed the
number of bins at four, with the locations of boundaries between the bins being
fitted by the model. Our new allometry using piecewise regressions led to predic-
tions of M with no apparent bias relative to D (Extended Data Fig. 4c). This new,
unbiased allometry gave the samequalitative results as our original, simple allometry
regarding the relationship betweenG andM: for all nine species,G increased in the
bin containing the largest trees, regardless of the allometry used (Extended Data
Fig. 5). We conclude that any bias associated with the minority of our species that
used the simple allometric equation form was unlikely to affect our broad conclu-
sion that G increases with tree size in a majority of tree species.
As a final assessment, we compared our results to those of a recent study of
E. regnans and S. sempervirens, in whichM andG had been calculated from inten-
sive measurements of aboveground portions of trees without the use of standard
allometric equations7. Specifically, in two consecutive years 36 trees of different
sizes and ageswere climbed, trunkdiameterswere systematicallymeasured at several
heights, branch diameters and lengths were measured (with subsets of foliage and
branches destructively sampled to determine mass relationships), wood densities
were determined and ring widths from increment cores were used to supplement
measured diameter growth increments. The authors used these measurements to
calculateM for each of the trees in each of the two consecutive years, and G as the
difference in M between the two years7. E. regnans and S. sempervirens are the
world’s tallest angiosperm and gymnosperm species, respectively, so the data set
was dominated by exceptionally large trees; most had M$ 20Mg, and M of some
individuals exceeded that of the most massive trees in our own data set (which
lacked E. regnans and S. sempervirens). We therefore compared E. regnans and
S. sempervirens to the 58 species in our data set that had at least one individual
withM$ 20Mg. Sample sizes for E. regnans and S. sempervirens—15 and 21 trees,
respectively—fell below our required$40 trees for fitting piecewise linear regres-
sions, so we simply plotted data points for individualE. regnans and S. sempervirens
along with the piecewise regressions that we had already fitted for our 58 compar-
ison species (Fig. 3).
As reported by ref. 7,G increasedwithM for bothE. regnans and S. sempervirens,
up to and including some of themostmassive individual trees on the Earth (Fig. 3).
Within the zone of overlapping M between the two data sets, G values for indi-
vidual E. regnans and S. sempervirens trees fell almost entirely within the ranges of
the piecewise regressions we had fitted for our 58 comparison species. We take
these observations as a further indication that our results, produced using standard
allometric equations, accurately reflect broad relationships between M and G.
Fitting log–log models. To model log(G) as a function of log(M), we used the
binning approach thatweused inourprimary analysis ofmass growth rate (described
earlier). However, in log-transforming growth we dropped trees withG#0. Because
negative growth rates become more extreme with increasing tree size, dropping
them could introduce a bias towards increasing growth rates. Log-transformation
additionally resulted in skewed growth rate residuals. Dropping trees with G# 0
caused several species to fall below our threshold sample size, reducing the total
number of species analysed to 381 (Extended Data Fig. 2).
Growth in the absence of competition. We obtained published equations for 41
North American and European species, in 46 species-site combinations, relating
species-specific tree diameter growth rates to trunk diameterD and to neighbour-
hood competition45–49. Setting neighbourhood competition to zero gave us equa-
tions describing estimated annual D growth as a function of D in the absence of
competition. Starting atD05 10 cm, we sequentially (1) calculated annualD growth
for a tree of size Dt, (2) added this amount to Dt to determine Dt1 1, (3) used an
appropriate taxon-specific allometric equation to calculate the associated tree
massesMt andMt11, and (iv) calculated tree mass growth rateGt of a tree of mass
Mt in the absence of competition as Mt1 12Mt. For each of the five species that
had separate growth analyses available from two different sites, we required that
mass growth rate increased continuously with tree size at both sites for the species
tobe considered tohave a continuously increasingmass growth rate.NorthAmerican
and European allometries were taken from refs 17 and 50, respectively, with pre-
ference given to allometric equations based on power functions of tree diameter,
large numbers of sampled trees, and trees spanning a broad range of diameters. For
the 47% of European species for which ref. 50 had no equations meeting our
criteria, we used the best-matched (by species or genus) equations from ref. 17.
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of an El Nin˜o dry season. J. Trop. Ecol. 20, 51–72 (2004).
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34. Chave, J. et al. Tree allometry and improved estimation of carbon stocks and
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38. Niklas, K. J. Plant allometry: is there a grand unifying theory? Biol. Rev. 79,
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ExtendedData Figure 1 | Summary ofmodel fits for treemass growth rates.
Bars show the percentage of species with mass growth rates that increase with
tree mass for each bin; black shading indicates percentage significant at
P# 0.05. Tree masses increase with bin number. a, Species fitted with one bin
(165 species); b, Species fitted with two bins (139 species); c, Species fitted with
three bins (56 species); and d, Species fitted with four bins (43 species).
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Extended Data Figure 2 | Log–log model fits of mass growth rates for 381
tree species, by continent. Trees with growth rates# 0were dropped from the
analysis, reducing the number of species meeting our threshold sample size
for analysis. a, Africa (33 species); b, Asia (123 species); c, Australasia
(22 species); d, Central and South America (73 species); e, Europe (41 species);
and f, North America (89 species). Trunk diameters are approximate values for
reference, based on the average diameters of trees of a given mass.
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Extended Data Figure 3 | Aboveground mass growth rates for 41 tree
species in the absence of competition. The ‘1’ or ‘2’ symbol preceding each
species code indicates, respectively, species with mass growth rates that
increased continuously with tree size or species with mass growth rates that
declined in the largest trees. Sources of the diameter growth equations used to
calculatemass growthwere: a, ref. 45;b, ref. 46; c, ref. 48;d, ref. 47; and e, ref. 49.
ABAM, Abies amabilis; ABBA, Abies balsamea; ABCO, Abies concolor; ABLA,
Abies lasiocarpa; ABMA, Abies magnifica; ACRU, Acer rubrum; ACSA, Acer
saccharum; BEAL, Betula alleghaniensis; BELE, Betula lenta; BEPA, Betula
papyrifera; CADE,Calocedrus decurrens; CASA,Castanea sativa; FAGR, Fagus
grandifolia; FASY, Fagus sylvatica; FRAM, Fraxinus americana; JUTH,
Juniperus thurifera; PIAB, Picea abies; PICO, Pinus contorta; PIHA, Pinus
halepensis; PIHY, Picea hybrid (a complex of Picea glauca, P. sitchensis and
P. engelmannii); PILA, Pinus lambertiana; PINI, Pinus nigra; PIPINA, Pinus
pinaster; PIPINE, Pinus pinea; PIRU, Picea rubens; PIST, Pinus strobus; PISY,
Pinus sylvestris; PIUN, Pinus uncinata; POBA, Populus balsamifera ssp.
trichocarpa; POTR, Populus tremuloides; PRSE, Prunus serotina; QUFA,
Quercus faginea; QUIL,Quercus ilex; QUPE, Quercus petraea; QUPY, Quercus
pyrenaica; QURO, Quercus robar; QURU, Quercus rubra; QUSU, Quercus
suber; THPL, Thuja plicata; TSCA, Tsuga canadensis; and TSHE, Tsuga
heterophylla.
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Extended Data Figure 4 | Residuals of predicted minus observed tree mass.
a, The allometric equation for moist tropical forests34—used for themajority of
tree species—shows no evident systematic bias in predicted aboveground dry
mass, M, relative to trunk diameter (n5 1,504 trees). b, In contrast, our
simplest form of allometric equation—used for 22% of our species and here
applied to nine temperate species—shows an apparent bias towards
overestimating M for large trees (n5 1,358 trees). c, New allometries that
we created for the nine temperate species removed the apparent bias in
predicted M.
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Extended Data Figure 5 | Estimated mass growth rates of the nine
temperate species of Extended Data Fig. 4. Growth was estimated using the
simplest form of allometric model [log(M)5 a1 blog(D)] (a) and our
allometric models fitted with piecewise linear regression (b). Regardless of the
allometric model form, all nine species show increasing G in the largest trees.
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