Ecology, 87(3), 2006, pp. 535–541
? 2006 by the Ecological Society of America
FUNDAMENTAL TRADE-OFFS GENERATING THE WORLDWIDE LEAF
BILL SHIPLEY,1,5MARTIN J. LECHOWICZ,2IAN WRIGHT,3AND PETER B. REICH4
1De ´partement de biologie, Universite ´ de Sherbrooke, Sherbrooke, Quebec J1K 2R1 Canada
2Department of Biology, McGill University, 1205 Avenue Dr. Penfield, Montreal, Quebec H3A 1B1 Canada
3Macquarie University, Department of Biological Sciences, Sydney 2109 Australia
4Department of Forest Resources, University of Minnesota, 1530 Cleveland Avenue North, St. Paul, Minnesota 55108 USA
leaf traits that affects global patterns of nutrient cycling and primary productivity and that
is used to calibrate vegetation–climate models. The correlation patterns are displayed by
species from the arctic to the tropics and are largely independent of growth form or phy-
logeny. This generality suggests that unidentified fundamental constraints control the return
of photosynthates on investments of nutrients and dry mass in leaves. Using novel graph
theoretic methods and structural equation modeling, we show that the relationships among
these variables can best be explained by assuming (1) a necessary trade-off between al-
location to structural tissues versus liquid phase processes and (2) an evolutionary trade-
off between leaf photosynthetic rates, construction costs, and leaf longevity.
Key words: comparative ecology; leaf life span; leaf longevity; leaf mass per area, LMA; leaf
nitrogen content; net photosynthetic rate; path analysis; specific leaf area, SLA; structural equations
modeling, SEM; vanishing tetrads.
Recent work has identified a worldwide ‘‘economic’’ spectrum of correlated
The results of natural selection in diverse environ-
ments, contingent on various physiochemical and phy-
logenetic constraints, are mirrored in the multivariate
patterns of correlation among heritable traits of func-
tional significance. Consider leaf form and function.
Given the incredible variation in leaf form and phys-
iological performance, given the deep phylogenetic or-
igin of this variation, and given the wide variety of
environments in which leaves operate, one might ex-
pect many different types of trade-offs involving leaf
form and function. If so, then the patterns of correlation
among functional leaf traits should vary greatly across
differing phylogenies and environments. This expec-
tation is contradicted by nature (Reich et al. 1997,
Wright et al. 2004). A single principal component cap-
tures 74% of the total variance in six key foliar traits
in the GLOPNET data set (Wright et al. 2004) which
involves 2548 species from 219 families representing
175 sites from the arctic to the tropics. Fully 82% of
the variation in the four variables of interest in the
present paper (maximum photosynthetic rate, leaf mass
Manuscript received 18 July 2005; revised 4 October 2005;
accepted 5 October 2005. Corresponding Editor: A. M. Ellison.
per area, leaf life span, and nitrogen content per mass)
is captured by a single principal component.
Why does this suite of leaf traits correlate so strongly
in such different species and environments? Wright et
al. (2004) suggest that it ‘‘reflects a mixture of direct
and indirect causal relationships between traits,’’ but
what is the nature of such relationships? It is not pos-
sible to answer this question in an interspecific context
through experimental manipulation. One cannot, for
example, either randomly assign, or directly manipu-
late, values of leaf mass per area (LMA) to widely
different species independently of their other leaf at-
tributes. An alternative approach, made possible by the
marriage of confirmatory structural equations modeling
and recent advances linking the structure of directed
acyclic graphs with probability distributions, is to de-
rive the predicted patterns of trait covariation implied
in different hypothetical causal processes and statisti-
cally compare these to the empirical patterns.
Here, we adopt this novel approach to study the re-
lationships among four key leaf attributes in the GLOP-
NET data set: maximum photosynthetic rate (Am), leaf
mass per area (LMA), nitrogen content per mass (Nm),
and leaf longevity (LL). We first test two alternative
path models that have been proposed in the literature.
We then apply two new exploratory algorithms based
on directed acyclic graphs to discover the correlational
BILL SHIPLEY ET AL.
Ecology, Vol. 87, No. 3
in the literature proposed to explain the interspecific patterns
of covariation among leaf mass per area (LMA), maximum
leaf photosynthetic rate on a mass basis (Am), leaf life span
(LL), and leaf nitrogen concentration per dry mass (Nm).Error
variances (e) are also shown. Both models are rejected at a
probability below 3 ? 10?4.
Directed acyclic graphs of two competing models
constraints on inter-relationships among traits that exist
in these data and, using these results, develop a new
hypothesis to explain the worldwide leaf economics
spectrum. Finally, we test this hypothesis using struc-
tural equations modeling.
METHODS AND MATERIALS
The GLOPNET data set contains 2496 observations
on 2020 different species, including 492 species for
which data were available for all four variables, the
rest having missing values for at least one of the four.
The combination Am–LL had the fewest observations
(511) and the average number of observations per var-
iable pair was 1128 (the complete data set is available
online).6To assure homogeneity of variance and line-
arity, all observed variables were transformed to their
natural logarithms. Structural equation models were fit
using the MPLUS program (Muthe ´n and Muthe ´n 1998–
2001) by maximizing the likelihood conditional on the
structural constraints implied by the model when com-
paring the model and the empirical covariance matri-
ces. Further details are found in the Appendix. Follow-
ing Little and Rubin (1987), we assumed that missing
values were ‘‘missing at random,’’ which is consistent
with the diverse decisions on what to measure made
by the many investigators whose results appear in the
compiled data. Other assumptions of the likelihood test
are approximate multivariate normality and linearity in
the trends between the variables after data transfor-
mation. Judging from normal quantile plots, we believe
that the data satisfy both assumptions.
Vanishing tetrads were evaluated using the TETRAD
program and tested using Wishart’s asymptotic deri-
vation for the sampling variance (Wishart 1928). The
exploratory path analysis assuming no latent variables
was done using the EPA2 program. Both programs are
described in Shipley (2000a:306). Details of the tetrad
representation theorem and the algorithm for explor-
atory path analysis are given in the Appendix.
Testing pre-existing structural hypotheses
We identified two different explanations in the lit-
erature for the patterns of covariation among these
traits (Fig. 1). Fig. 1a is taken from the interpretation
given by Wright et al. (2004): ‘‘. . . the linkage of high
AMwith high NMis in large part the result of a direct
causal relationship (Field and Mooney 1986) (i.e., Nm
→ Am). Similarly, long LL requires the robustness and
low palatability (including chemical defenses) associ-
ated with high LMA (i.e., LMA → LL). More indi-
rectly, high Amtends to be associated with short LL
because it requires high Nmand/or low LMA (i.e., LMA
↔ Nm), which increase leaf vulnerability to herbivory
and physical hazards, and because high Amdrives fast
growth, rapidly shading older leaves, leading them to
senesce once their resources become more valuable
when transferred to better-lit newer foliage (i.e., Am→
LL).’’ The second explanation (Fig. 1b, involving Am,
Nm, and LMA) is based on the path model of Meziane
and Shipley (2001) plus the preceding expectation in-
volving LL. Using structural equation modeling, we
tested both models representing the hypothesized caus-
al origins of the functional relationships among the
traits. Both models were unequivocally rejected (model
1a, ?2? 12.825, df ? 1, P ? 0.0003; model 1b, ?2?
209.37, df ? 2, P ? 1 ? 10?15), suggesting that they
do not adequately explain the covariation among these
Although many additional models are conceivable,
is there any testable ordering of the four variables that
fits the data without invoking any additional variables
influencing the patterns of correlation? If such an or-
dering exists, then there will be at least one path model
March 2006 537
TRADE-OFFS PRODUCING LEAF ECONOMIC TRAITS
with positive degrees of freedom that fits the data (Shi-
pley 2000a). We used the CI (Pearl 2000) and FCI
(Spirtes et al. 2000) algorithms, which provide an ex-
haustive search over the space of potential orderings,
to obtain potential path models with positive degrees
of freedom that then were each tested using the d-sep
test of Shipley (2000b, 2003) at a significance level of
P ? 0.05 and using Pearson (partial) correlation co-
efficients; details of these tests are in the Appendix.
This search was based only on the 492 observations
for which all four variables were measured. No such
fitting model was found. This is strong statistical ev-
idence that the interspecific patterns of correlation
among these four leaf traits are generated by one or
more variables not considered in this initial analysis
(i.e., latent variables). Even stronger statistical evi-
dence for the influence of latent variables is obtained
using the tetrad representation theorem, which specifies
the necessary and sufficient conditions for detecting
the influence of at least one unmeasured (latent) var-
iable on the relationships among measured variables.
Of the three possible tetrad equations involving the four
measured leaf traits in our analyses, only one tetrad
equation is not significantly different from zero:
?? ? ??
ln(LL),ln(N ) ln(LMA),ln(A )
(z ? 1.404, P ? 0.16), and there is only one pair of
measured variables (Amand LL) common to the two
nonzero tetrads. This means that all causal paths linking
every pair of variables except for LL and Ampass
through the same latent variable.
ln(LL),ln(LMA) ln(N ),ln(A )
A new hypothesis for the worldwide
leaf economics spectrum
Given this evidence for a latent variable, we propose
an alternate explanation for the patterns of correlation
among the four measured foliar traits that involves (1)
unmeasured variation in cell size and cell wall thick-
ness and (2) selection on leaf longevity to maximize
plant carbon gain rather than leaf-level carbon gain.
Variation in cell size and cell wall thickness.—We
divide the volume of a leaf (VL) into three compart-
ments: the volume occupied by air spaces (Va), the
volume occupied by cell walls (Vw), and the volume
bounded by the cell membranes of living cells (Vc).
This third compartment consists mostly of liquid. Here,
the cell wall is not included in the cell volume. This
is a slight modification of the model of Roderick (Rod-
erick et al. 1999a, b). The ratio Vc/Vwis determined by
the average size of a living cell in the leaf relative to
the thickness of the average cell wall.
Nitrogen per dry mass.—Leaf nitrogen is found pri-
marily within the cell, not in the cell wall (Hikosaka
2004). Let the average mass of nitrogen per cell volume
be n ¯. The total mass of nitrogen per leaf (N) is therefore
N ? n ¯Vcwith strict equality if all nitrogen is in the
cell. Let the average density of the tissue mass be d.
The amount of nitrogen per dry mass (Nm) in a leaf
with a tissue mass (Mt) that is found both within the
cell (Mc) and in the cell wall (Mw) is
M ? M
M ? M
If the cell does not contain large qualities of nonstruc-
tural carbohydrates then
M k M
where d (specific gravity) is relatively constant among
species at 1.5 (Desch 1973). Since nitrogen per liquid
volume is only half as variable as Nmacross species
(Roderick et al. 1999b), n ¯ will be even less variable,
implying that most of the interspecific variation in Nm
is due to variation in cell size vs. cell wall thickness
(i.e., Vc/Vw→ Nm).
Net photosynthesis per dry mass.—Carbon fixation
in the Calvin cycle occurs within the lumen (liquid
phase) of the chloroplast. Furthermore, chloroplast
number per cell increases with cell size in interspecific
comparisons (Pyankov et al. 1999, Pyke 1999). There-
fore, if a ¯ is the average rate of net carbon fixation per
cell volume and A is the total net carbon fixation per
leaf, then the rate of net photosynthesis per dry mass
M ? M
(i.e., Vc/Vw→ Am).
Leaf mass per area (LMA).—
start with specific leaf area (SLA) and note that LMA
? SLA?1. Since the total volume of a leaf (VL) is the
product of projected leaf surface area (S) and the av-
erage leaf thickness (T),
For simplicity, we
M ? M
T(M ? M )
V ? V ? V
and LMA will respond in the same way but with op-
posite signs; i.e., it will decrease with an increasing
ratio Vc/Vw(i.e., Vc/Vw→ LMA).
Optimal leaf longevity.—Kikuzawa’s (1991) model
predicts that if natural selection optimizes plant carbon
fixation per unit time over the life of a leaf, then the
optimal leaf longevity should increase with decreasing
maximum net photosynthetic rate of the leaf once ex-
pansion has terminated (i.e., LL → Am), and should
increase with increasing leaf construction cost (i.e., CC
→ Am). The construction cost is the total amount of
resources (ultimately energy) that the plant must invest
to construct the leaf. Let the construction cost (usually
quantified in terms of glucose-equivalents expended)
needed to absorb, fix, transport or biochemically ma-
BILL SHIPLEY ET AL.
Ecology, Vol. 87, No. 3
cell [Vc], the cell wall [Vw], and air spaces [Va], lamina thickness [T], mass density of the cell wall [d], average mass of
nitrogen [n ¯], other elements within the cell [e ¯i] per cell volume, average mass of elements within the cell wall per cell wall
volume [e ¯j], average rate of net carbon fixation per cell volume [a ¯], and construction cost of the leaf per dry mass [?m]). (b)
Equivalent predicted relationships involving only observed variables and a single latent (L) in the form of a structural equation
(a) Theoretical relationships between the observed variables in Fig. 1 and unmeasured variables (volumes of the
nipulate each atom of element i in the newly expanded
leaf be ?i. If the leaf requires a total mass of this element
equal to Eito construct the leaf until it becomes au-
totrophic, then the total cost to the plant with respect
to this element is ?iEiand the total construction cost
of the leaf is ? ? ??iEi. Since most carbon except for
nonstructural carbohydrates is found in the cell wall,
and construction costs do not include such nonstruc-
tural reserves, the total mass of carbon is approximately
c ¯Vwwhere c ¯ is the carbon mass per cell wall volume
and is quite constant across species (Roderick et al.
1999b). The total mass of each of the mineral elements
(ei) that are found primarily inside the cell is e ¯iVc, where
e ¯iis the mass of element i per cell volume. Separating
elements that are located primarily in the cell (indexed
by i) or the cell wall (indexed by j), the cost to construct
the leaf is
? ?? e ¯ V ?
? e ¯ V .
Leaf construction cost per unit dry mass is therefore
M ? M
? e ¯
?? e ¯ .
Thus, the construction cost per unit dry mass of com-
pounds within the cell will increase with the ratio of
cell volume to cell wall volume, while the cost of com-
pounds in the cell wall will be independent of this ratio
(i.e., Vc/Vw→ ?m).
Together, the variation in cell size and cell wall thick-
ness and Kikuzawa’s theory of optimal leaf longevity
predict the trade-offs shown in Fig. 2a. The resulting
structural equation model is shown in Fig. 2b. The
dotted line will be absent if the concentration of nitro-
gen per cell volume is constant (or too small to be
detected). Assuming the interspecific variation in n ¯ to
be negligible, we fit the GLOPNET data to Fig. 2b
using ln-transformed variables and without the dotted
line. Since we did not possess information on (Vc/Vw),
this variable is latent, and its scale was fixed by con-
TRADE-OFFS PRODUCING LEAF ECONOMIC TRAITS
variance of the latent (L), and the r2values of the observed variables are shown, with the standard errors of estimates in
The structural equation model of Fig. 2b including the fitted values. Path coefficients, the residual variances, the
straining the path coefficient (L → Nm) to unity as spec-
ified by the theory. The resulting model coefficients
are shown in Fig. 3; this model fits the data remarkably
well (?2? 0.766, df ? 1, P ? 0.38) despite a very
large sample size and the signs of all coefficients are
in the predicted direction; note that this analysis is
based on the full data set with missing values not only
the 492 full observations. These coefficients represent
the average responses over the full range of species
and sites represented in the data set. Since regression
analyses of bivariate pairs of these variables have de-
tected site-specific differences in the slopes (Wright et
al. 2005), it is likely that the path coefficients will also
show such variation. The residual variances of all var-
iables are significantly different from zero, indicating
that there are additional unmeasured causes of each,
and some of this residual variation might be due to
These results provide support for the theoretical ex-
planation developed above but do not unambiguously
identify the latent variable as (Vc/Vw); this would re-
quire direct measurement of this ratio. However, we
did have a subset of 82 observations of Australian ev-
ergreen species for which we have measures of leaf
water and dry mass. Assuming that leaf water mass is
approximately equal to cell volume and that leaf dry
mass is approximately equal cell wall mass (both ap-
proximations being poorer as leaves accumulate non-
structural carbohydrates), the ratio of leaf water to leaf
dry mass (Wm) is approximately equal to (Vc/Vw). We
therefore tested the model shown in Fig. 2b using these
new data plus the water mass:dry mass ratio as an
additional indicator variable whose path coefficientwas
fixed at 1.0 since it should scale isomorphically with
the latent variable, as described above; the path coef-
ficient from the latent variable to Nmwas also fixed to
unity as done in Fig. 2b. The result is shown in Fig.
4. The model without the Nm→ LL path is rejected (?2
? 15.245, df ? 4, P ? 0.004), but the model including
this path fits the data well (?2? 4.080, df ? 3, P ?
0.39). Comparing Fig. 4b with Fig. 3, the main struc-
tural difference of this site-specific model is that leaf
longevity primarily trades off with Nmrather than Am.
The strength of the correlation between Wmand the
latent is lower than we had expected (r ? 0.66). This
might reflect both that a nonnegligible proportion of
the dry mass (e.g., nonstructural carbohydrates) was
not in the cell wall, and also that a nonnegligible pro-
portion of the leaf water can be found in the cell wall
matrix (Berry and Roderick 2005). The rejection of the
model in Fig. 4a in favor of the one in Fig. 4b might
indicate either that the general explanation developed
above requires revision or simply that these Australian
species are unusual.
Despite a very large sample size and resulting sta-
tistical power to detect errors, we have produced a
structural equation model that fits the data exception-
ally well. This model suggests that the worldwide leaf
economics spectrum has a surprisingly simple under-
lying structure that is generated by two general trade-
offs. The first trade-off involves size–number relation-
ships of living cells within the leaf (Pyankov et al.
1999, Niinemets 2001) and variation in cell wall thick-
ness; this will be affected by the proportion of different
types of tissues in the leaf. This first trade-off generates
the broad interspecific coordination between Am, LMA,
and Nmand also determines the main interspecific dif-
ferences in construction cost per dry mass of a leaf.
This, in combination with the general tendency for pho-
tosynthetic rate to decrease with leaf age that is the
basis for Kikuzawa’s model, generates the second
trade-off between Amand leaf longevity; namely that
an increased Amin a young leaf will decrease leaf lon-
BILL SHIPLEY ET AL.
Ecology, Vol. 87, No. 3
values. Path coefficients, residual variances, the variance of the latent (L), and the r2values of the observed variables are
shown, with the standard errors of estimates in parentheses. Path coefficients not significantly different from zero are shown
by broken lines.
The structural equation model of Fig. 2b, plus the mass of water per dry leaf mass (Wm) including the fitted
It is perhaps surprising that models that posit direct
effects between physiological variables such as net
photosynthetic rate and leaf nitrogen content (cf. Fig.
1) were rejected in favor of a model (Fig. 3) that posits
an indirect relationship mediated by size–number re-
lationships of living cells within the leaf. An expla-
nation is suggested by recognizing that these physio-
logical processes occur at different scales (Marks and
Lechowicz 2005). At the scale of a single chloroplast,
the physiological mechanisms directly linking the pho-
tosynthetic enzymes and pigments to carbon fixation
are well established. However, the GLOPNETvariables
are measured at the scale of an entire leaf; at this scale
interspecific variation in the relationship between ni-
trogen amount and carbon fixation can arise both
through interspecific differences in the functioning of
a single chloroplast and through interspecific variation
in the size and number of cells and chloroplasts in
different leaves. Presumably, variation in these leaf-
level variables is dominated by how differing numbers
and volumes of cells and cell constituents are put to-
gether to form a leaf, while the physiological processes
at the subcellular level are much more constant.
Our model does not require that all species have the
same values for the path coefficients in all environ-
ments, only that the topological linkages between the
variables (the way they link together) be the same. The
estimated path coefficients are average values that cap-
ture the general tendency in the data; it is likely that
separate intraspecific models would detect significant
differences in the quantitative values of the path co-
efficients. Wright et al. (2005) detected significant, but
quite small, levels of heterogeneity in slopes among
these variables between sites although much of the var-
iability could be explained by differences in sample
size. The degree to which species can escape from the
two general trade-offs represented in our model is
quantified by the residual variances. Since these resid-
ual variances were all significantly different from zero,
this means that there are additional causes for each of
these variables but that these additional causes are spe-
cies and site specific. Understanding the origin of these
residual variances is critical since, hidden within it, are
the signals of small-scale differential adaptation to par-
This research was financially supported by the Natural Sci-
ences and Engineering Research Council (NSERC) of Canada
and by the National Science Foundation of the United States.
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A description of the statistical analysis, including details of testing a structural equation model, identifying latent variables
via vanishing tetrad differences, and the CI algorithm for exploratory path analysis (Ecological Archives E087-029-A1).