Oikos 000: 001–007, 2013
doi: 10.1111/j.1600-0706.2013.00436.x
© 2013 The Authors. Oikos © 2013 Nordic Society Oikos
Subject Editor: Thorsten Wiegand. Accepted 17 March 2013
A general combined model to describe tree-diameter distributions
within subtropical and temperate forest communities
Jiangshan Lai, David A. Coomes, Xiaojun Du, Chang-fu Hsieh, I-Fang Sun, Wei-Chun Chao,
Xiangcheng Mi, Haibao Ren, Xugao Wang, Zhanqing Hao and Keping Ma
J. Lai, X. Du, X. Mi, H. Ren and K. Ma ([email protected]), State Key Laboratory of Vegetation and Environmental Change,
Inst. of Botany, Chinese Academy of Sciences, CN-100093 Beijing. – D. A. Coomes, Forest Ecology and Conservation Group,
Dept of Plant Sciences, Univ. of Cambridge, Downing Street, Cambridge, CB2 3EA, UK. – C. Hsieh, Inst. of Ecology and Evolutionary Biology,
National Taiwan Univ., Taipei. – I.-F. Sun, Dept of Natural Resources and Environmental Studies, National Dong Hwa Univ., Hualien.
– W.-C. Chao, Dept of Forestry and Natural Resources, National Chiayi Univ., Chiayi. – X. Wang and Z. Hao,
State Key Laboratory of Forest and Soil Ecology, Inst. of Applied Ecology, Chinese Academy of Sciences, CN-110164 Shenyang.
The size distribution of trees in natural forests is a fundamental attribute of forest structure. Previous attempts to model
tree size distributions using simple functions (such as power or Weibull functions) have had limited success, typically
overestimating the number of large stems observed. We describe a model which assumes that the dominant mortality
process is asymmetric competition when trees are smaller, and size-independent processes (e.g. disturbance) when trees
are larger. This combination of processes leads to a size distribution which takes the form of a power distribution in the
small tree phase and a Weibull distribution in the large tree phase. Analyses of data from four large-scale ( 24 ha each)
subtropical and temperate forest plots totalling 99 ha and approximately 0.4 million trees provide support for this model
in two respects: (a) the combined function provided unbiased predictions and (b) power-law functions fitted to small trees
had exponents that deviated from the universal exponent of 22 predicted by metabolic scaling theory, gradually decreasing from subtropical evergreen to temperate deciduous forests along the latitudinal gradient.
The size-frequency distribution of trees is a fundamental
attribute of forest structure that is well correlated with other
features including forest biomass, carbon storage and gross
production (Niklas et al. 2003, Yen et al. 2010, Stephenson
et al. 2011). Although it is widely recognised that tree
abundance declines with increasing tree size, there is still
uncertainty about the mechanistic basis for this relationship
and whether there is a universal mathematical formula to
describe this pattern (Enquist et al. 1998, Bokma 2004,
Brown et al. 2004, Reynolds and Ford 2005).
Various probability density functions have been used
to describe the size distributions of forests (Meyer 1952,
Bailey and Dell 1973, Hafley and Schreuder 1977,
Maltamo et al. 1995, 2000), but most are phenomenolo­
gical models which lack a mechanistic basis (Enquist et al.
2009). For example, the Weibull distribution is a flexible
distribution commonly used to fit tree-diameter distributions (Little 1983, Zutter et al. 1986, Cao 2004, Bullock
and Burkhart 2005, Palahi et al. 2006, Coomes and Allen
2007), but its theoretical basis has received little attention
and it is usually applied in a phenomenological way.
Muller-Landau et al. (2006b) showed that demographic
process modelling (Kohyama et al. 2003) gives rise to
Weibull distributions when growth is a power-law function
of size, mortality is constant and the forest in dynamic equilibrium. Moreover, although the Weibull distribution is
shown to fit well for a range of tropical forests (MullerLandau et al. 2006b), it performs less well in temperate
forests (Wang et al. 2009), highlighting the need to develop
a model that is general along latitudinal gradients.
The –2 power rule of size distributions which forms part
of metabolic scaling theory of ecology (MST) (Enquist et al.
1998, 2009, Enquist and Niklas 2001) has a simple mechanistic basis derived from an understanding of plant physio­
logy and geometry (Midgley 2001), but the fact that most
empirical size distributions deviate from this rule suggests
that more complex process models are required (Coomes
et al. 2003, Coomes 2006, Muller-Landau et al. 2006b,
Coomes and Allen 2007, Wang et al. 2009, Simini et al.
2010, Anfodillo et al. 2013). Numerous factors may contribute to variation in tree size distributions among and within
sites, including species composition, site-specific resource
availability, community age and disturbance history (Condit
et al. 1998, Coomes et al. 2003, Niklas et al. 2003, MullerLandau et al. 2006b, Enquist et al. 2009).
Here we argue that the tree-size distributions spanning
the entire range of tree sizes cannot be predicted by a simple
theoretical model based on consideration of a single process,
EV-1
as an ontogenetic shift in dominant ecological processes
determines a change in the functional form of the size distribution in different growth stages (Franklin et al. 1987,
Coomes et al. 2003). We assume intense size-asymmetric
competition is the major cause of stem mortality at small
stem diameters, and size-independent processes are important at large stem diameters, showing that in combination
these processes can lead to a power distribution in small
tree phase and a Weibull distribution in large tree phase.
Data from four large forest dynamics plots from subtropical
to temperate forests along a latitudinal gradient in east Asia
are used to test the combined power-Weibull model. The
model successfully predicts the tree size distribution in these
forests. Our study not only provides the alternative model
to predict general features of size distribution in different
forests, but also proposes changes in exponents for the power
function rather than the universal exponent of 22 predicted
by metabolic scaling theory.
Theory and methods
Theoretical model
As in earlier work, we assume that forests are in demographic
equilibrium, and that there is no recruiment limitation
or distinction among tree species (Enquist and Niklas
2001, Muller-Landau et al. 2006b, Enquist et al. 2009). We
discuss these assumptions later.
Tree diameter distribution in the small tree phase
Previous studies have shown that size-asymmetic competition can lead to power-law distributions in tree size due to
‘thinning rules’ whereby trees in a population partition
space in a geometric manner and energy optimization (Yoda
et al. 1963, Adler 1996, Begon et al. 1996, Enquist and
Niklas 2001, Enquist et al. 2009, Simini et al. 2010,
Anfodillo et al. 2013). For example, Enquist and Niklas
(2001) developed an individual-based simulator in which
trees grew, produced seeds and dispersed those seeds according to a simple set of rules, and died when competition for
light from taller neighbours starved them of resources: they
found that the size distribution at dynamic equilibrium was
well-approximated by a power function with a 22 exponent.
However, there has been controversy spanning several
decades over the precise slope of self-thinning relationships,
and whether similar mechanisms drive patterns in agricultural monocultures and mixed-aged mixed-species forests
(Deng et al. 2012). Furthermore, little consideration has
been given to the shift in mortality process away from
asymmetric competition when tree reach the large tree phase
(Coomes et al. 2003). It is therefore desirable to explore
whether the observed power function for the small tree
phase deviates from a ‘22 scaling rule’ and whether it
changes systematically along a latitudinal gradient from
evergreen broadleaved to deciduous broadleaved forest.
The tree-size distribution is a frequency distribution and
is thus characterized by a probability density function. In
cases where inventory data are collected only for trees above
a minimum diameter Dmin (e.g. 1 cm dbh), a left-truncated
distribution function is necessary. Here, we first predict that
EV-2
the stem density in the small tree phase should scale with
tree diameter. Therefore, the power function has fixed lower
and upper limits. The power-law probability density function with fixed lower and upper limits for the small
tree phase is expressed as a Truncated Pareto distribution
(White et al. 2008):
1θ 1 θ
f ( D )  (1 θ )(Dt1θ  Dmin
) D (1)
where θ is the exponent and Dt is the threshold diameter
above which a tree is categorised as being in the large tree
phase rather than the small tree phase.
Diameter distribution in the large tree phase
Assuming that size-independent processes kill large trees
such that the probability of death is size invariant, then
the annual mortality rate is constant:
m(D)  c
(2)
By allometric theory, diameter growth rate is assumed to
take a simple power-function form (Calder 1984, Enquist
et al. 1999, Economo et al. 2005, Russo et al. 2007):
dD
(3)
 rD α
dt
where r and a are constant and D is stem diameter. Because
we are not considering the distinction among different
tree species, we assume all trees follow the same growth function. Following the derivation of demographic equilibrium
theory (Muller-Landau et al. 2006b) under these conditions
the diameter distribution of the large tree phase will take the
form of a left-truncated Weibull function with shape para­
meter a and scale parameter b  c/r(1 2 a):
g ( D )
f (D ) 
β(1 α )Dα exp(βD1α )
exp(βDt1α )
(4)
Therefore, our second prediction is that tree size distribution
in the large tree phase follows a Weibull distribution.
A model for describing the entire size distribution is a
combined power-Weibull function, comprised of Eq. 1 and
4. There is a switch between functions at some threshold
diameter Dt representing the size at which trees grow into the
large tree phase and beyond the influences of asymmetric
competition.
Data analyses
We tested the model using tree data from four large forest
plots in east Asia (Table 1): our analyses are based on nearly
400 000 individuals across 99 ha of land. The four sites
are distributed in different climate zones along a latitudinal
gradient (23°N to 42°N), ranging from subtropical evergreen to cool temperate deciduous forests, and are representative of vegetation types in these areas. All plots have been
established and censused following the plot protocols of the
Centre for Tropical Forest Science (CTFS) (Condit 1998).
All stems with diameter at breast height (dbh)  1 cm have
been mapped, measured (with a precision of 0.1 cm), iden­
tified, and tagged.
Table 1. Characteristics of the four large forest dynamics plots used in this study.
Sites
Lianhuachih
Gutianshan
Baotianman
Changbaishan
Area
(ha)
Latitude
(°N)
Longitude
(°E)
Climate
Rainfall (mm year21)
No. of
trees
Census
date
Ref
25
24
25
25
23.9
29.3
33.5
42.3
120.9
118.1
111.9
128.0
south subtropical
north subtropical
warm temperate
cool temperate
2285
1964
886
700
153268
140087
59527
36904
2008
2005
2011
2004
(Lin et al. 2011)
(Lai et al. 2009)
†
(Wang et al. 2009)
†Temperate deciduous broadleaved forest consisting of 118 species (73 genera and 38 families) and dominated by Quercus aliena var.
acutiserrata (Fagaceae) in Neixiang County, Henan Province in China.
The best approach for determining the threshold dia­
meter Dt at which there is a switch between two functions
is a challenge. Simini et al. (2010) developed a finite sizescaling method for determining the range of tree sizes within
which a power law holds, based on the assumption that the
diameter probability distribution has a sharp drop off in
probability as diameter approaches Dt. However, for three of
our sites (Fig. 1) the diameter probability distribution does
not show any indication of a sharp drop off – in fact the
slope of the curve become less negative in the Weibull versus
the power-law phase. Therefore, we used maximum likelihood estimation to determine the most suitable threshold
diameter Dt. Maximum likelihoods of the combined
power-Weibull model were calculated for a range of possible
Dt values, ranging from 3 to 30 cm in 1 cm increments
(Supplementary material Appendix A1 A) and the diameter
with the greatest maximum likelihood was selected at the
threshold.
The size distribution of individuals with diameters
from 1 cm to Dt was modelled using Eq. 1 (Dmin  1 cm)
while the distribution for trees greater than Dt was modelled
using Eq. 4. Maximum likelihood estimation (MLE), the
preferred method for estimating parameters of frequency
distributions (White et al. 2008), was used to estimate
empirical parameters q, a and b in our combined powerWeibull model. Since the power function has fixed upper
and lower limits here, the special truncated Pareto distribution is recommended to unbiased estimate such exponents
by White et al. (2008):
ln D 
1θ
1θ
1 Dmax ln Dmax  Dmin ln Dmin

1θ
1θ
1 θ
Dmax  Dmin
(5)
where D is the diameter of each individual, Dmin and Dmax
are the minimum and maximum diameter values (1 cm and
Dt, respectively, in combined model). The eq. 5 to obtain
exponent q cannot be solved analytically, but must be
solved with numerical methods (White et al. 2008). The lefttruncated Weibull distribution (Eq. 4), as described by
(Muller-Landau et al. 2006b) was fitted to trees  Dt diameter. Confidence intervals of 95% on parameters were
obtained from 1000 bootstraps. A common and simple
approach to evaluate our models is to plot predicted versus
observed bin data values and compare against the 1:1 line
(Smith and Rose 1995, Mesple et al. 1996). We excluded
data for individuals whose diameters were recorded as 1.0
cm to avoid the influence of inconsistencies in the definitions of this smallest size class.
We used Akaike’s information criterion (AIC), predicted
versus observed plot and residuals plot to compare the goodness of fit of our combined power-Weibull model against
using pure power or Weibull functions to describe the entire
distribution. AICs were calculated as 22log-likelihood  2k,
where k is the number of parameters. All calculations in
this paper were conducted using R statistical language
(R Development Core Team) (see Supplementary material
Appendix A1 E for all R codes used in this study).
Results
The tree-size distributions of the four forest plots spanning
a latitudinal gradient in east Asia are shown in Fig. 1. A
common feature of the diameter distributions for small tree
phase is linearity on log-log plots, and downward curvature
for large tree phase. The log-log graphs also indicate that
neither pure power functions (green lines in Fig. 1 and their
parameters in the Supplementary material Appendix A1 D)
nor pure Weibull functions (blue lines in Fig. 1) accurately
describe the whole tree diameter distribution, except in the
case of Lianhuachih plot which approximates to a pure
Weibull distribution on visualization (Fig. 1, also see scatter
plots of predicted vs observed value in the Supplementary
material Appendix A1 B). Deviations from pure power
functions or pure Weibull functions always occur in the
large tree phase, particularly in the three higher latitude
plots, Gutianshan, Baotianman and Changbaishan forest.
The log-likelihood values for our combined powerWeibull model, as a function of the evaluated threshold
diameter, Dt, first increased then decreased in all four plots
(Supplementary material Appendix A1 A). The peaks are
obvious and easily determine the most suitable threshold
diameter Dt. In contrast, Dt varied widely among forests
(Table 2): Lianhuachih plot, the southernmost site in our
study, Dt  7 cm, is much less than those in other three forest plots (14 cm for Gutianshan, 16 cm for Baotianman and
12 cm for Changbaishan). We will discuss this pattern later.
The choice of threshold diameter Dt appears appropriate
for all four plots (see Fig. 1 in particular the point of departure from linearity in the three higher latitude plots, Gutianshan, Baotianman and Changbaishan forests). The size
distri­butions of small stems were closely described by a power
function at all sites, while the size distributions of large
stem phase were well described by a Weibull distribution
(red lines in Fig. 1). All scatter plots of predicted versus
observed binned data (Supplementary material Appendix A1
B) also indicated our model provides accurate fits to tree
size distributions in both subtropical evergreen and temperate deciduous forests. Although our combined model with
a threshold diameter Dt seems more complex than
pure power or Weibull distribution, the improvement in
goodness-of-fit is substantial: the combined model is always
EV-3
EV-4
5e-05
1e-03
5e-02
1e-05
1e-03
1e-01
2
Baotianman
Lianhuachih
5
10
20
50
100
2
Tree stem diameter, D (cm)
5
Changbaishan
Gutianshan
10
20
50
100
Figure 1. Observed tree size distributions (points  1 cm size-class bins) for four large forest plots in east Asia spanning a latitude gradient, and the predictions of a combined power-Weibull
function (red lines, dashed line for threshold diameter Dt), power function (green lines) and Weibull function (blue lines).
Density of tree
Table 2. The fit of the combined Power-Weibull model to the diameter distribution data for four large forest plots in east Asia (ordered from
subtropical to cool-temperate).
Sites
Lianhuachih
Gutianshan
Baotianman
Changbaishan
Threshold
diameter
Dt (cm)
Tree
density
(ha21) (Dt)
Power function
exponent (q)
(95% CI)
Tree
density
(ha21) (Dt)
Weibull function
shape (a)
(95% CI)
Weibull function
scale (b)
(95% CI)
7
14
16
12
5101
4915
2036
1057
1.76 (1.75, 1.77)
1.73 (1.72, 1.74)
1.68 (1.66, 1.70)
1.65 (1.63, 1.67)
1011
532
317
377
0.47 (0.45, 0.49)
20.46 (20.50, 20.40)
20.78 (20.85, 20.72)
20.40 (20.44, 20.36)
0.77 (0.70, 0.85)
0.015 (0.013, 0.019)
0.0025 (0.0019, 0.0032)
0.0089 (0.0074, 0.0108)
strongly supported statistically (DAIC  500 in all sites)
(Table 3) and has smaller absolute value of residuals over
the entire range of tree sizes (Supplementary material
Appendix A1 C).
The absolute value of exponents of the small tree phase
power function declined systematically from 1.76 to 1.65
along the latitudinal gradient (Fig. 1, Table 2). Although
the power-law function provides an excellent description of
size distributions in the small tree phase, the exponents
deviate significantly from the 22 value proposed by metabolic scaling theory.
Discussion
The combined power-Weibull model described in this
paper has a solid mechanistic basis – building upon the
extensive self-thinning literature by including the effects of
disturbance on large trees – and provides accurate fits to all
our datasets. Power and Weibull distribution are commonly
used to describe size distributions in forests, but neither
proved adequate for describing size distributions across
the entire range of diameters measured in the east Asian
sites, particularly in the case of temperate forests.
Our results support the growing body of literature
demonstrating that there is no universal exponent of 22,
even though the power-law function is shown to describe
the size distribution of the small tree phase very closely
(Coomes et al. 2003, Niklas et al. 2003, Muller-Landau
et al. 2006b, Coomes and Allen 2007, Simini et al. 2010,
Anfodillo et al. 2013). The observation that the power-law
exponent becomes less negative from subtropical evergreen
to temperate deciduous forests with increasing latitude in
east Asia is consistent with the arguments put forward by
Niklas et al. (2003), that exponents will decline if there
are systematic declines in stem density. This situation is
Table 3. AIC comparisons of the combined power-Weibull function
with pure power and Weibull functions fitted to the whole tree –
diameter distribution at each site. ΔAIC is the magnitude the focus
AIC subtracting the minimal AIC.
DAIC value
Sites
Lianhuachih
Gutianshan
Baotianman
Changbaishan
Combined
model
Pure power
function
Pure Weibull
function
0
0
0
0
12771.0
12300.8
18133.8
24352.9
775.6
7609.8
14734.2
20790.1
consistent with a reduction in tree density in the small tree
phase, from 5101 ha21 to 1057 ha21 along the latitudinal
gradient represented by our plots (Table 2). However, it is
difficult to quantify the relationship between stand density
and exponent, because many other factors influence the
power law exponent (White et al. 2008, Clauset et al. 2009).
We note that Enquist et al. (2009) supported their argument for a 22 scaling rule by analysing stem-diameter data
from several large plots, and find exponent close to 22 for
different censuses. However, their analysis is based on the
entire range of tree sizes, and ignore the fact that the powerlaw is inadequate for this purpose. Our results also support
the concept of finite-size scaling (Maritan et al. 1996),
which argues pure power law behaviour can hold only
over a limited range of sizes (Simini et al. 2010, Anfodillo
et al. 2013).
Yet we know of no that studies have recognised the size
distribution at the large tree phase follow an independent
Weibull distribution rather than the small tree phase.
We not only used our data to show such pattern, more
importantly, our derive process also provides a mechanistic
foundation for interpreting the Weibull distribution. Our
derivation of the Weibull distribution for large tree phase
is consistent with the approach of Muller-Landau et al.
(2006b), showing that this distribution arises when
growth is described by a power function and mortality is
size independent, under the framework of demographic
equilibrium theory (Kohyama et al. 2003). It is important
to bear in mind that the demographic equilibrium theory
does not in itself predict any particular size distribution; it
simply provides an approach to calculate the expected size
distribution given growth and mortality and assuming
equilibrium. There is abundant evidence that tree diameter
growth scales with diameter size, but the growth-diameter
scaling exponents varies substantially among species and
environmental conditions (Muller-Landau et al. 2006a,
Price et al. 2007, Russo et al. 2007, 2008, Coomes and
Allen 2009, Coomes et al. 2011, Stark et al. 2011), and we
are not yet in a position to predict how such variation will
affect parameters of the Weibull distribution for tree size
distribution in different forests. Furthermore, it is wellknown that exogenous disturbances (e.g. insect infestations,
diseases and catastrophic weather events) are a major source
of large trees mortality in natural forests (Kanzaki and Yoda
1986, Wells et al. 2001, Batista and Platt 2003, Woods
2004), and that these events are highly unpredictable and
stochastic. In the absence of human activities, wind disturbance may be one of the most important natural disturbances in east Asian forests (Mabry et al. 1998, Lin et al.
EV-5
2003, Nakajima et al. 2009). Whilst some previous studies
have shown mortality rates to be near constant across size
classes in the large tree phase (Coomes et al. 2003) others
have found size-dependent mortality (Davies 2001, MullerLandau et al. 2006a). Further analyses are required to test
whether the mortality rates in the large tree phase in east
Asian forests are size invariant, and to explore the
implications of relaxing assumptions on the size distribution of the large trees. We recognise that another explanation for the Weibull distribution evident in the large tree
phase is finite size-scaling in the small tree phase (Simini
et al. 2010, Anfodillo et al. 2013); more research is needed
to critically evaluate these alternative explanations.
Note though that our southernmost site, Lianhuachih, is
located on Taiwan island. The forest has a much smaller
threshold diameter Dt than other three, perhaps because the
local area is especially exposed to typhoon disturbance from
the Pacific Ocean (Lin et al. 2003). Following our argument
that Dt is the size at which trees become less influenced by
asymmetric competition and more influenced by disturbance, frequent and strong typhoon damage in the
Lianhuachih area, which affects trees of all sizes, may provide
an explanation for the low Dt value.
Our model predictions match well with the empirical
data, but this does not mean our assumptions are necessarily
accurate. Indeed, it is known that many forests are not in
dynamic equilibrium but are undergoing ‘stand development’ or succession following catastrophic disturbance,
with competitive interactions driving changes in local stand
structure over time (Coomes and Allen 2007, Coomes et al.
2012). Moreover, a multitude of processes may contribute
to tree mortality, aside from competition and disturbance
factors (Franklin et al. 1987). It is possible that our study
plots are so large that the influences of multiple processes
are conflated into a single signal. At the 25-ha scale, our
combined power-Weibull is well supported for four forest
types spanning a wide latitudinal range. This suggests our
assumptions capture fundamental aspects of the biology
of forest communities. This paper focusses explores tree
size distributions in terms of stem diameter, but we recognised that other aspect of size, such as tree height and
crown volume, are more directly influential in the processes
driving tree size distribution in forest communities (Strigul
et al. 2008, Simini et al. 2010, Anfodillo et al. 2013).
Further work is required to collect tree height and crown
sizes information and refine our model.
Understanding and predicting tree-size distributions is
valuable in both basic and applied ecology. Practically
speaking, information on tree size distribution is essential for
calculating forest stand yields and allocating efforts in tending and protecting forests. Our study not only recognises
the differentiation of size distribution between small and
large tree phases, but also provides a mechanistic model to
predict such features of size distribution in different forests.
Our results may contribute to improving the accuracy of
forest aboveground carbon estimation by integrating two
size distribution function (Stephenson et al. 2011); this may
be particularly useful for predicting the number of large
diameter trees in forests; these account for a major portion
of the biomass but are greatly overestimated by simple treesize distribution functions.
EV-6
Acknowledgements – We thank many dedicated people for the skilled
labour and generous funding that generated the forest dynamics
data sets which provide the empirical basis for this research. This
work was motivated by discussions with Fangliang He. We also
thank Mrs. Lily van Eeden for checking the draft manuscript. The
data analyses reported in this study were supported by the Natural
Science Foundation of China projects (31200403, 31270496 and
31070554).
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Supplementary material (available as Appendix oik-00436
at  www.oikosoffice.lu.se/appendix ). Appendix A1.
EV-7