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(American Journal of Botany. 2008;95:549-557.) doi: 10.3732/ajb.0800034 © 2008 Botanical Society of America, Inc. |
What's this? |
Ecology |
Department of Plant Biology, Cornell University, Ithaca, New York 14853 USA
Received for publication 11 October 2007. Accepted for publication 25 February 2008.
ABSTRACT
Research indicates that increases in total leaf area (AT) may fail to keep pace with increases in total leaf mass (ML) across plants differing in size (e.g., as measured by stem diameter, D). This "diminishing returns" hypothesis predicts that the scaling exponent for AT vs. ML will be less than one and that the exponent for specific leaf mass (i.e., AT / ML) vs. D will be negative. These predictions were examined using data from 46 plants ranging between 0.125 cm 
D 0.485 m across 25 woody dicot species. Standardized major axis slopes were used to quantify scaling exponents and random effects models were used to quantify species and size effects on the numerical values of exponents. The exponents for AT vs. ML and AT / ML vs. D differed among species and different species groupings. In general, the exponent for AT vs. ML was less than one and the exponent for AT / ML vs. D was negative, as predicted. However, random effects models indicated that species effects overshadowed size effects, although size effects were statistically significant. The diminishing returns hypothesis therefore receives statistical support, i.e., although the numerical values of exponents are "species-dependent," they are less than unity, as predicted by theory.
Key Words: allometry • functional foliar traits • leaf area • leaf economics • leaf mass • leaf number • random effects models • scaling relationships • specific leaf area
What is the "dividend" gained in total leaf area as the "investment" in total leaf dry mass increases, and how do total leaf area and total leaf mass change as plants grow in overall size? These questions are fundamental when considering the "economics" of light-harvesting, both at the level of the individual plant and at the level of comparisons among diverse species differing in their functional foliar traits. It is surprising, therefore, that despite extensive research into the foliar allometry, little is actually known about the scaling relationships among total leaf number, area, and mass at the whole plant level, particularly for tree-sized specimens.
Nevertheless, there are some suggestions that these scaling relationships may be suboptimal for gains in leaf area with increasing investing in leaf dry mass. For example, analysis of an extensive database for mature leaf size (>5000 measurements on 1943 species) indicates that the mean lamina surface area (
) increases as the 0.979 power of the mean lamina dry mass (
) across fern and angiosperm (vine, graminoid, forb, shrub, and tree) species (Niklas et al., 2007
). Taken at face value, this 
0.979 relationship suggests that any increase in the total leaf mass investment will, on average, result in "diminishing returns" in total leaf area. Specifically, if total leaf area per plant (AT) equals 
NT (where NT is total leaf number) and if total leaf mass (ML) equals NT, (where and are true and not estimated means), it follows mathematically from aL mL0.979 that AT mL0.979 NT < ML. If shown to be valid empirically, this diminishing returns hypothesis has important ecological and evolutionary implications. For example, disproportionately smaller gains in total leaf surface area with increasing leaf mass or overall plant size would increase the "cost" of harvesting light, which would help to explain why annual growth rates in dry mass per plant fail to keep pace with increases in total body mass across otherwise ecologically and phyletically diverse species (Enquist et al., 1998; Enquist and Niklas, 2002; Niklas and Enquist, 2001, 2002). Such an insight may be particularly timely and important because attempts to explain this phenomenon based on theory continue to be criticized for biological and mathematical reasons (see Kozlowski and Konarzewski, 2004).
Unfortunately, however, the postulate that gains in total leaf area fail to keep pace with increases in total leaf dry mass investments reveals nothing about the actual scaling of AL with respect to ML other than it is likely to be governed by an exponent (denoted here by 
) that is numerically less than unity (i.e., AT ML < 1.0), because any attempt to derive the numerical value of based on theory is frustrated mathematically by what has been called the "fallacy of the averages." Specifically, if two or more variables manifest covariance across species or among conspecifics differing in size, their multiplicand can differ significantly from the actual values of the quantities sought after (Wagner, 1969; Welsh et al., 1988). It is a common practice to compute total leaf area (or total leaf mass) by multiplying mean leaf area (or mean leaf mass) by total leaf number (e.g., White, 1983a, 1983b; Sterck and Bongers, 2001; Poorter et al., 2006; Dekker et al., 2007). However, this approach is valid if and only if mean leaf area and mean leaf mass are computed based on an extensive sampling of all leaves. Setting aside the practical limitations of measuring and accurately, particularly for large plants bearing thousands of leaves, there is another important caveat, i.e., there is no reason a priori to assume that individual plants maintain the same covariance among , , and NT as they increase in size. Likewise, prior research has identified significant levels of interspecific covariance among leaf size, number, and a host of other important foliar, functional traits (e.g., White, 1983a, 1983b; Midgley et al., 1995; Shipley, 1995; Reich et al., 1997, 2003; Wright et al., 2006). Indeed, foliar trait covariance extends into deep geological time (see Royer et al., 2007). Accordingly, detailed empirical information about the allometric relationships among , , and NT for conspecifics differing in size (or across diverse species differing in mature leaf size) is required to determine the scaling relationship for AT with respect to changes in ML.
Attempts to determine this scaling relationship empirically are equally frustrated due to an insufficient number of published paired direct measurements for tree-sized plants. As noted, many studies report the scaling relationships between mean leaf area and mean leaf mass based on random samples of leaves (e.g., Whittaker et al., 1974; Burton et al., 1991; Shipley, 1995; Wright et al., 2001, 2006; Niklas et al., 2007). However, most are based on measurements made on twigs or isolated branches rather than extensive samples drawn at the whole plant level (e.g., White, 1983a, 1983b; Brouat et al., 1998). Equally frustrating, some authors measure AT and ML directly for tree-sized plants but fail to report the raw data. For example, in an elegant study of tree-level hydraulics, Yang and Tyree (1994) measured AT and ML directly, but they did not report the actual value of ML for the trees examined. Indeed, few studies are as comprehensive as those of Turrell (1934, 1961) who reported the total leaf area of a single Catalpa speciosa tree measuring 0.43 m in diameter (Turrell, 1934) and the total leaf area and mass for four Citrus sinensis trees between 9.14 cm and 0.27 m in diameter (Turrell, 1961).
This paper tests the hypothesis of diminishing returns by examining the scaling exponent governing the relationship between total leaf area and total leaf mass, which is predicted to be numerically less than one, and the scaling exponent for the relationship between specific leaf area (total leaf area divided by total leaf mass; AT / ML) and stem diameter, which is predicted to be negative. Paired and direct measurements of total leaf area and total dry mass from 46 specimens of tree species (representing 22 dicot genera and 25 species) ranging between 0.125 cm and 0.485 m in diameter are used for this purpose (Table 1).
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, 2004, 2006; Hui and Jackson, 2007) and as a function of differences in the size-range sampled during analyses, even within the same species (see Brouat et al., 1998, fig. 4; Niklas, 1994, fig. 3.32; Sterck and Bongers, 2001, fig. 8). The values of exponents also depend on the statistical protocols used to determine them (Sokal and Rohlf, 1981; Niklas, 1994; Warton et al., 2006; Hui and Jackson, 2007), and they can differ significantly even among closely related species (Prior et al., 2003; Weiner, 2004; Niklas 2006). These caveats highlight the possibility that the numerical values for scaling exponents can be sampling artifacts reflecting the selection of species or the sizes of conspecifics examined during the course of any study. It is apparent, for example, that different samples of the same hypothetical species can yield very different exponents for the interspecific scaling of total leaf area vs. total leaf mass and leaf specific area vs. stem diameter depending on the sizes of conspecifics and the types of species samples (Fig. 1).
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) and random effects models (also known as restricted or residual maximum likelihood; see Wolfinger, 1993, 1996; Singer, 1998) analysis were used to evaluate respectively the numerical values of scaling exponents and to determine whether these values differed at different levels of comparison. Limitations imposed by using these protocols are discussed. MATERIALS AND METHODS
Data from 46 specimens (representing 25 species and 22 genera) within the 0.001 m D 0.485 m size range were used in our analyses (Table 1). These data include those published for one specimen of Catalpa speciosa and four specimens of Citrus sinensis (Turrell, 1934, 1961). Among the species investigated, six species had pinnate (vs. simple) leaves. Tree as opposed to herbaceous species were selected for study because D, NT, AT, or ML span many orders of magnitude across tree species, which fosters the statistical determination of allometric (scaling) exponents.
With the exception of data for C. speciosa and C. sinensis and two Acer rubrum saplings and one Quercus alba sapling, all specimens grew in the Ithaca, New York area. Each was selected on the basis of three criteria: each (1) had to be free standing (to preclude possible effects of shading on total leaf number NT and ML), (2) show no obvious signs of major branch or leaf damage (which could bias estimates of ML and AT), and each (3) had to have a single trunk (to provide unambiguous measurements of D). Destructive sampling was required to measure total dry leaf mass and area, limiting access to specimens growing on private property and, in turn, the species available for study.
Basal stem cross-sectional area was measured with the aid of a metric metal ruler and used to compute D. Stem diameter as opposed to cross-sectional area was selected to measure overall plant size because of the availability of large data sets relating D to annual plant growth, total leaf dry mass, and a number of other variables of biological interest (see Niklas, 1994, 2004, 2006; Niklas and Enquist, 2001). Total leaf area was determined for specimens with 50 leaves by photocopying all leaves after their removal from stems, cutting out lamina images with a razor, and calculating AT using a linear regression curve for photocopy paper area vs. weight (N = 25, r2 = 0.999), after correcting for image-distortion and checking for paper thickness uniformity.
For plants with >50 leaves, all leaves were removed from each specimen and sorted into size classes based on lamina length measured with a metric ruler. The number of length size classes ranged between 8 and 12 (average size class number = 8) and increased with increasing overall plant size. In the majority of cases, the mean surface area of 15–20 leaves drawn from each size class (
) was subsequently determined (using the weight vs. area linear regression; see preceding paragraph) and multiplied by the number of leaves in each size class (ni). Total leaf area per plant was computed as AT = 
ni. Leaf dry mass was measured using the same sampling technique because Turrell (1934, 1961) showed unequivocally that AL 
NT and ML NT if mean values are computed from small samples of leaves randomly drawn from all size classes (an observation confirmed during our study). However, in the majority of cases, total leaf mass was measured directly for all specimens by weighing leaves at constant weight after air drying. In the case of large specimens (i.e., NT 
2000), leaves were packaged in coolers and air dried in patches (by spreading them across a large floor). Petiole or rachis surface areas and dry mass were not included in measurements of AT and ML.
Preliminary analyses of regression residuals showed that all bivariate relationships among D, NT, ML, and AT were log-log linear; all subsequent statistical analyses used log10-transformed data. Standardized major axis (also known as reduced major axis) slopes and intercepts ( and log β, respectively) were used to quantify the allometric relationships among D, NT, AT, and ML, i.e., all regression curves took the general form log Y1 = β ± log Y2, where Y1 and Y2 denote dependent variables. Slopes and intercepts and their respective 95% confidence intervals (CIs) were computed using the software package Standardised Major Axis Tests and Routines (SMATR), version 2 (website http://www.bio.mq.edu.au/ecology/SMATR/) (see Warton and Weber, 2002; Warton et al., 2006). SMATR analyses were checked using closed-form formulas for the 95% CIs of and log β (Jolicoeur, 1990). In all cases, comparisons of closed-form derived CIs agreed with SMATR results. These protocols were applied to (1) the entire data set (N = 25 species and 46 specimens), (2) all species represented by two or more specimens (N = 15 species and 36 specimens), (3) all species represented by two or more specimens with the exception of A. rubrum (N = 14 species and 31 specimens), and (4) all specimens of species for which we sampled three or more plants.
Random effects models (REM; also known as restricted or residual maximum likelihood; see Singer, 1998) analysis were used to evaluate the effect of species selection on the scaling relationship between specific leaf area (AT / ML) and plant size (measured either as D or NT). In these models, "species" was designated to have a random effect, whereas plant "size" was designated as a fixed effect. In an effort to examine the potential mechanism(s) that might account for intraspecific changes in total leaf area, mass, and specific leaf area (i.e., total leaf area / total leaf mass), the leaves of four A. rubrum conspecifics differing in size were extensively sampled to determine their leaf size-frequency distributions.
All statistical protocols other than standardized major axis regression analyses were performed using the statistical software package JMP version 7.0 (SAS Institute, Cary, North Carolina, USA).
RESULTS
Across the 46 specimens, each of the variables of interest ranged over three or more orders of magnitude (Figs. 2–4
). Regression analyses indicated that the slopes and elevations (i.e., and log β, respectively) of the log-log regression curves for all bivariate plots did not statistically differ between those species with compound leaves and those with simple leaves (6 vs. 19 species, respectively; see Table 1). For example, SMATR analysis of AT vs. ML gave = 0.880 (96% CIs = 0.710, 1.05) for plants with compound leaves (N = 7 specimens, r2 = 0.967) and = 0.749 (95% CIs = 0.662, 0.835) for those with simple leaves (N = 39 specimens, r2 = 0.922). The data from all 46 specimens therefore were pooled and subsequently subdivided to determine the log-log scaling relationships among all the variables of interest and to examine species and plant size statistical effects.
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= 1.08, 95% CIs = 1.06, 1.11; Table 2). Regression analysis of the data for Q. alba was suspect, however, because the distribution of the three specimens in our data set was uneven. The total leaf area and total leaf mass of the two larger plants were similar and differed considerably from the smallest plant (see Fig. 2B species no. 8).
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Random effects model (REM) analyses of these data also identified scaling exponents that were less than one for the AT vs. ML relationship for each of the five different species groupings; these REM exponents (which are not mathematically equivalent to the standardized major axis regression exponents given in Table 2) varied between 0.755 and 0.785. However, REM analyses also identified statistically significant species effects in addition to size effects (Table 3). Within the A. rubrum and Q. alba species grouping, species contributed 18.2% to the total variance as opposed to differences in the size of total leaf mass among individuals within each of these two species. The addition of C. sinensis to the A. rubrum/Q. alba group decreased the size effect by 38.3% (from 81.8% to 43.5%). Across the remaining three species groupings, species affiliation contributed between 64.5% and 67.4% to the total variance. REM analysis nevertheless indicated that the size effect was statistically significant within each of the five regressions (i.e., P < 0.0001).
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Specific leaf area vs. plant size (AT / ML vs. D)
Across all species, specific leaf area (AT / ML) was observed to numerically decline with increasing basal stem diameter (D) (Fig. 4A). Within individual species represented by three or more specimens differing in D (i.e., Acer rubrum, Citrus sinensis, and Quercus alba; see Table 1) and across different groupings of different species, standardized major axis regression analyses consistently identified negative scaling exponents for the AT / ML vs. D scaling relationship (Table 5). With the exception of the data for Citrus sinensis, the numerical values of the AT / ML vs. D scaling exponents were statistically indistinguishable For example, the scaling exponents for A. rubrum and Q. alba were statistically indistinguishable as were those for the A. rubrum, C. sinensis, and Q. alba grouping from all other species groupings. However, the numerical value of the exponent for the C. sinensis AT / ML vs. D scaling relationship based on four specimens was statistically suspect because the smallest tree among the four (D = 0.09 m; data from Turrell, 1934) was a statistical outlier, which improved the regression analysis when removed (Table 5). Similarly, the AT / ML vs. D scaling exponent for Q. alba was suspect because two among the three AT / ML data points had similarly high values for similarly large D and because the third data point represented a very small plant (i.e., the regression for Q. alba was "leveraged" and thus uninformative (Fig. 4B). Also, grouping the data for A. rubrum and Q. alba was statistically not legitimate because the regression curves for these two species did not share the same "elevations" (i.e., their log β-values differed substantially).
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Random effects model (REM) analyses of these data also identified negative scaling exponents for the AT / ML vs. D relationship for each of the five different species groupings; these REM exponents (which are not mathematically equivalent to the standardized major axis regression exponents given in Table 5) varied between –0.517 and –0.365. However, REM analyses also identified statistically significant species effects in addition to size effects (Table 6). Within the A. rubrum and Q. alba species grouping, species effects contributed 52.2% to the total variance as opposed to differences in the sizes among individuals within each of these two species. The addition of C. sinensis to the A. rubrum/ Q. alba group decreased the size effect by 33.7% (from 47.8% to 14.1%). Across the remaining three species groupings, species affiliation contributed between 71.4% and 74.9% to the total variance. Nevertheless, REM analysis indicated that the size effect was statistically significant within each of the five regressions (i.e., P < 0.0001).
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Acer rubrum foliar size-frequency distributions
In an effort to explore how AT / ML and D (and NT) are negatively correlated at the level of individual species, we examined the size-frequency distributions of leaves from four of the five A. rubrum conspecifics representing the extreme ends of the size spectrum available for this species.
Among these specimens, AT / ML frequency distributions differed significantly both in their shape and sizes (Fig. 5). The AT / ML frequency distributions of the two smallest conspecifics (D = 0.004 m and 0.0087 m; NT = 5 and 9 leaves, respectively) occupied the largest size categories of the distributions of the two largest conspecifics (D = 0.083 m and 0.31 m; NT = 4175 and 16890 leaves, respectively). Therefore, the leaves on these small, juvenile plants were, on average, larger in area but lighter in dry mass than those produced by the two larger plants (Fig. 5 A insert). A comparison between the shapes of the frequency distributions of the two largest conspecifics indicated that the frequency distribution of the largest plant was "shifted" respectively to size bins left of the size distribution of the second largest specimen (Fig. 5 A, B). Although environmental effects cannot be ruled out as causative agents, these trends were interpreted to indicate that the AT / ML frequency distribution of A. rubrum may vary ontogenetically during the transition from plant establishment to maturity. In addition, among all four conspecifics, mean leaf dry mass () was significantly and positively correlated with D (r2 = 0.889, P = 0.016), whereas mean leaf area () was not (r2 = 0.038, P = 0.755) such that differences in the AT / ML frequency distributions among these plants resulted from differences in the dry mass investments in different leaf size categories.
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For the majority of species and species groupings, our analyses are consistent with two predictions of the diminishing returns hypothesis. The scaling exponents for the relationship for total leaf area vs. total leaf mass are generally less than one, and the exponents for the relationship between specific leaf area vs. stem diameter are negative, as predicted by the hypothesis. However, this "generality" is suspect statistically because random effects model analyses indicate that the variance component estimates for species effects are significantly larger than those of size effects, which are nevertheless statistically significant. The numerical values of all scaling exponents of species groupings therefore are dependent on the species composition of the data sets used. This species-dependence is clearly evident when the exponents of individual species are examined and compared, e.g., the scaling exponent governing the relationship between total leaf area and total leaf mass for Citrus sinensis is positive and has 95% confidence intervals that exclude values less than one (see Table 2).
However, in support of the hypothesis, we note that previous studies relating one or more of the variables of interest examined in this study report similar scaling relationships to those presented here. For example, total leaf dry mass has been reported to scale as the square of basal stem diameter (Niklas, 2004); total leaf area is reported to scale less than the square of stem diameter, or, if an isometric relationship is reported, the 95% CIs of the slopes include numerical values well below unity (Turrell, 1961; Yang and Tyree; 1994; Brouat et al., 1998). Similarly, in their study of Malaysian rain forest species, Thomas and Bazzaz (1999) found that the specific leaf areas of adult trees had a significant negative correlation with overall plant size (measured as maximum height), whereas the AT / ML of saplings was higher than that of older conspecifics.
The only relevant analogous data set known to us that deviates from the interspecific trends reported here is that of White (1983a, 1983b) for which SMATR analysis indicates that the AT of first year tree shoots scales as the 2.04 power of D (N = 20, r2 = 0.936). The 95% CIs for this exponent include values less than 2.0 (i.e., 95% CIs = 1.79, 2.32), but the trend evident using White’s data differs from that observed for the tree species examined by us. This dissimilarity may reflect differences in the allometry of nonwoody vs. woody shoots or the fact that White (1983a, 1983b) reports mean values for AT and D rather than raw paired measurements (and is thus suspect in light of "the fallacy of averages"). We believe the latter is more likely.
The hypothesis of diminishing returns is also supported by a variety of other studies that adduce the existence of trade-offs among key foliar functional traits (e.g., Shipley, 1995; Reich et al., 1997, 2003; Price and Enquist, 2007). For example, Niinemets et al. (2006) have shown that the dry mass fraction invested in the mechanical tissues of leaves increases disproportionately with increases in total leaf mass across different species (presumably at the expense of lamina surface area); Niklas et al. (2007) and subsequently Milla and Reich (2007) have shown independently that, at the level of individual mature leaves, the scaling exponent governing the relationship between leaf surface area and dry mass is less than one; and Pearcy et al. (2005) have shown that increases in leaf size or number require disproportionately larger biomechanical investments in the construction of subtending stems (which presumably would be reflected in increases in total leaf mass).
The differences in the numerical values of scaling exponents reported by us and other workers for different species are important for at least two reasons. First, the hypothesis of diminishing returns does not apply to every species, and second, it does not a priori suggest negative consequences on the performance of physiological processes linked directly or indirectly to total lamina surface areas at the level of individual species. Our data clearly show that some species fail to manifest diminishing returns in how total leaf area scales with respect to total leaf mass or how specific leaf area scales with respect to basal stem diameter, i.e., some species "violate" the hypothesis’ predictions. Likewise, even for those species that appear to conform to the predictions of this hypothesis, size-dependent changes in metabolic rates, leaf longevity, leaf size frequency distributions (e.g., favoring increases in the smaller leaf size classes), or other foliar functional traits could individually or collectively compensate for smaller gains in total leaf area with increasing total leaf mass or plant size. Indeed, changes in leaf-size frequency distributions and mean leaf dry mass with increasing plant size that could mitigate the potentially negative effects of total leaf area "diminishing returns" are evident in our data as well as those reported by other workers. Finally, it must not escape attention that differences in the elevations of log-log linear regression curves (i.e., the numerical values of log β) for total leaf area vs. total leaf mass among individual species can profoundly affect the ability to efficiently harvest light. All other things being equal, species with regression curves that have high elevations would have an advantage over those with lower elevations because they would have larger absolute values of total leaf area.
Nevertheless, there is reasonably strong circumstantial evidence that compensatory mechanisms such as these are inadequate at one or more levels across many species. For example, among otherwise very diverse species, total annual growth in dry mass (GT) is reported to scale as the 3/4 power of total dry body mass (MT), albeit isometrically with respect to total leaf mass (i.e., GT MT3/4 and GT ML) such that ML MT3/4 (see Niklas and Enquist, 2001; Niklas, 2004; Enquist and Niklas, 2002). These relationships are interesting from a purely theoretical perspective when juxtaposed with those reported here. For example, if AT ML < 1.0 and ML MT3/4 are true, on average, across different species differing in mature leaf size, it follows theoretically that the scaling exponent for AT vs. MT must be less than 3/4. Likewise, if GT ML and AT ML < 1.0 are true across species, the scaling relationship for GT vs. AT is expected to be larger than one. We did not measure GT for the specimens examined in this study. However, analysis of 20 of the 46 specimens indicates that the scaling exponent for AT vs. MT equals 0.58 (r2 = 0.916), which is significantly less than 3/4. In this way, the hypothesis of diminishing returns may shed light on how key foliar functional traits change with plant size and influence the capacity to harvest light (and thus growth in size), which have thus far been largely explored theoretically mostly in terms of metabolic models for organismal energetics (West et al., 1997; Enquist et al., 1998; Niklas and Enquist, 2001, 2002; Enquist and Niklas, 2002) that have been heavily criticized for a variety of reasons (Kozlowski and Konarzewski, 2004; Li et al., 2005).
Although a mechanistic explanation for diminishing returns remains speculative, our analyses indicate that, in the case of some species (e.g., Acer rubrum), mean leaf mass increases with increasing plant size, whereas mean leaf area remains largely unaffected, which results is an ontogenetic shift from high to lower specific leaf areas in the frequency distributions of leaves. This phenomenology, which results in an ontogenetic shift from high to lower specific leaf areas in the frequency distributions of leaves (see Fig. 5), may reflect differences in the amount of mechanical tissues in leaves differing in location within the canopy as well as changes in the spatial distribution and number of leaves exposed to differing ambient light intensities as a canopy grows in size (e.g., changes in the number of sun vs. shade leaves). Extensive studies are required to examine this hypothesis (as well as more global theories attempting to provide a fundamental explanation for diverse scaling phenomena among plants) both at the level of individual species and across different species. The challenge before us is to build sufficiently large and robust data bases, particularly those in which individual species are represented by large numbers of conspecifics differing in size. This task is essential in light of the sensitivity of the scaling exponents governing the relationships to intra- and interspecific variation resulting from the effects of local environmental conditions. For the moment, however, the notion that increases in overall plant body size come at some metabolic and energetic cost seems incontrovertible in light of the evidence accumulated thus far.
FOOTNOTES
1 The authors thank two anonymous reviewers and M. Christianson (AJB Associate Editor) for suggestions to improve an earlier version of this paper, F. Vermeylen (Cornell Statistical Consulting Unit) for providing tutorials in the use of random effects model analyses, and the many student volunteers who over the years helped to measure the lengths of over 233000 leaves. Support from the College of Agriculture and Life Sciences, Cornell University, is gratefully acknowledged. 
2 Author for correspondence (e-mail: kjn2{at}cornell.edu)
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50 m2/ kg). (B) AT/ MLfrequency distribution for a conspecific with D= 0.83 m (mean AT/ ML= 17.95 m2/ kg).