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(American Journal of Botany. 2003;90:250-254.)
© 2003 Botanical Society of America, Inc.

Ecology

Reexamination of a canonical model for plant organ biomass partitioning1

Karl J. Niklas2

Department of Plant Biology, Cornell University, Ithaca, New York 14853-5908 USA

Received for publication May 14, 2002. Accepted for publication August 13, 2002.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 LITERATURE CITED
 
A previously proposed "canonical" model for the scaling relations among leaf, stem, and root biomass (ML, MS, and MR, respectively) asserts that the proportional relations ML {propto} MS3/4 {propto} MR3/4 and MS {propto} MR hold across seed plant species. This model is scrutinized by determining whether the scaling relations between ML, MS, and MR vs. basal stem diameter DS and between ML, MS, and MR vs. plant height h are logically consistent with previously predicted scaling exponents. For example, if ML is observed to scale as the 2-power of DS and the model asserts that ML {propto} MS3/4, then MS must scale as the 8/3-power of DS if the model is valid. Using a large data base for species with self-supporting stems, statistical support was found for most such comparisons between predicted and observed scaling relationships. However, this judgement is predicated on (1) the assertion that the scaling exponents for MR with respect to DS (or h) are numerically "deflated" due to a systematic underestimate of fine and small root biomass and (2) the stringent protocol used to calculate the 95% confidence intervals of scaling exponents, which favors rejection of the model. In light of these features, the "canonical" model is logically consistent with the new scaling relations reported here. Therefore, the model is judged valid within the context of this evaluation.

Key Words: allometry • biomass allocation • organ biomass • plant height • scaling relations • stem diameter


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 LITERATURE CITED
 
An analytical approach to how the body mass of vascular plants is partitioned among the three vegetative organ types has identified a model for this partitioning pattern at the level of the individual plant (Niklas and Enquist, 2001 , 2002a , b ; Enquist and Niklas, 2002 ). This model asserts a "canonical" pattern in the sense that the scaling exponents describing vegetative organ biomass relations are expected to hold true, on average, across species regardless of ecological conditions. Specifically, standing leaf biomass ML is expected to scale as the 3/4-power of stem biomass MS and as the 3/4-power of root biomass MR such that MS {propto} MR (Niklas and Enquist, 2001 , 2002a , b ; Enquist and Niklas, 2002 ).

The scaling exponents predicted by this model comply remarkably well with those observed using a large database compiled by Cannell (1982) for the organ biomass relations among species with self-supporting stems (Enquist and Niklas, 2002 ; Niklas and Enquist, 2002a , b ). Nonetheless, the model rests on a number of as yet untested assumptions regarding the scaling of average stem (and root) diameter (and length). For example, the model explicitly assumes that total stem biomass scales isometrically with respect to the product of basal stem area and total stem length, i.e., MS {propto} DS2 LS (see Enquist and Niklas, 2002 ). Thus, the model must be approached with skepticism until its basic assumptions survive additional and repeated empirical scrutiny.

One test of the model is the extent to which the scaling exponents it predicts for vegetative organ biomass partitioning comply with those observed for the scaling of organ biomass with respect to either basal stem diameter or plant height. Logically, all such scaling relations must be internally consistent if the model is valid. For example, if leaf biomass is observed to scale as the {alpha}-power of stem diameter and if, as the model canonically asserts, leaf biomass scales as the 3/4-power of stem biomass, then it follows that stem biomass must scale as the {alpha}4/3-power of stem diameter, i.e., if ML {propto} DS{alpha} and ML {propto} MS3/4, then MS {propto} DS{alpha}4/3. Likewise, if leaf biomass scales as the {alpha}-power of plant height, then stem biomass must scale as the {alpha}4/3-power of height just as height must scale isometrically with respect to stem diameter, i.e., if ML {propto} h{alpha} and ML {propto} MS3/4, then MS {propto} h{alpha}4/3; if ML {propto} DS{alpha} and ML {propto} h{alpha}, then DS {propto} h.

In this paper, I examine the extent to which the model's predicted scaling exponents comply with those observed (1) for stem diameter vs. leaf, stem, or root biomass and (2) for stem diameter vs. leaf, stem, or root biomass. The Cannell database is used for this purpose because it provides information for average basal stem diameter and plant height, as well as organ biomass for the same community or site across a broad spectrum of woody dicot and conifer species, as well as a limited number of tall bamboo and palm species with self-supporting stems.

The analyses presented here provide reasonable but not unequivocal support for the "canonical" biomass partitioning model. For example, across the species represented in the Cannell database, leaf biomass scales as the 1.99-power of stem diameter, i.e., ML {propto} DS1.99. Since the model posits ML {propto} MS3/4 across species, it follows that stem biomass is expected to scale as the 2.65-power of stem diameter. Using the Cannell database, the observed scaling exponent for this relationship is 2.55. At issue is whether 2.55 is sufficiently statistically close to 2.65 to accept the assertion that ML {propto} MS3/4 holds across vascular plant species. This concern and others are discussed in the context of the internal consistency of all observed scaling relations and the manner in which the confidence limits of scaling exponents are calculated.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 LITERATURE CITED
 
The Cannell database
The Cannell (1982) compendium of approximately 600 published reports on standing organ biomass was examined to select data sets for monospecific stands or communities of mixed species providing the following information: number of plants per 1.0 ha (plant density), average plant height, total basal stem cross-sectional area, and the standing (dry) mass of leaves, stems, or roots. A total of 169 such communities (reflecting 695 observations reflecting different plant densities, water or light conditions, fertilization or thinning regimes, etc., for individual communities) was identified. Among the 169 communities, 465 were monotypic and predominantly even-aged stands (Table 1). All of the data were from tree-sized dicot, conifer, and monocot species with self-supporting stems. Across all pairwise comparisons, the taxonomic composition was biased in favor of conifer species, especially in terms of monotypic communities (Table 1).


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  		Table 1. Number, geographic distribution, and composition of Cannell (1982) data sets used to determine functional relations between standing leaf, stem, or root biomass vs. plant height (h) or basal stem diameter (DS) (see Table 2)

 
The standing biomass of leaves, stems, and roots per "average" plant in each community (and for each observation per community) was computed using the quotient of total community standing organ biomass and plant density. Average basal stem diameter was computed using the quotient of total community basal stem area and plant density. Values for average plant height were taken directly from Cannell (1982) .

A version of the Cannell compendium is provided online at www.sciencemag.org/cgi/content/full/295/559/1517/DC1; a copy of the data sets used in this study is available on request.

Statistical analyses
Model II (reduced major axis, RMA) regression analysis was used to determine scaling exponents ({alpha}RMA) and allometric constants (ßRMA) empirically for log10-transformed data using the formulas log Y2 = log ßOLS + {alpha}OLS log Y1, {alpha}RMA = {alpha}OLS/r, and log ßRMA = log Y2 – {alpha}RMA log Y1, where {alpha}OLS is the ordinary least squares (OLS) regression exponent, ßOLS is the OLS Y-intercept, r is the OLS correlation coefficient, and Y denotes the mean value of variable Y (see Sokal and Rohlf, 1981 ). This regression procedure is recommended when the variables of interest are biologically interdependent, subject to unknown measurement error, and when functional rather than predictive relationships are sought (see Sokal and Rohlf, 1981 ; Harvey and Mace, 1982 ; Peters, 1983 ; Rayner, 1985 ; McArdle, 1988 ; Niklas, 1994a ). The 95% confidence intervals for {alpha}RMA were computed using the formula CIOLS/r, where 95% CIOLS are the 95% confidence intervals of {alpha}OLS. The sample size N varied across pairwise comparisons because some authors did not report the biomass of one or two vegetative organs (see Table 2).


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  		Table 2. Statistics of pairwise Model Type II (reduced major axis) regression analyses of standing leaf, stem, and root biomass (ML, MS, and MR, respectively) with respect to basal stem diameter and plant height (DS and h, respectively) across all species and within the angiosperm and conifer data sets redacted from the Cannell (1982) compendium. In all cases, P < 0.0001

 
Two other statistical protocols were used to determine the scaling exponents (and their 95% confidence intervals) for the allometric relations reported here, i.e., eigenvectors of the principal component of the variance-covariance matrix (PC) and Bartlett's three-group method (BGM) (for numerical protocols; see Sokal and Rholf, 1981 ). The results of these analyses are not reported, because they did not lead to different conclusions based on the results of Model II regression analyses, i.e., the same predictions of the biomass-partitioning model were either accepted or rejected regardless of the type of regression analysis used.

Evaluating the model
The predictions of the "canonical" biomass partitioning model were evaluated by means of an "if X and Y, then Z" matrix, where X denotes a biomass scaling relation asserted by the model (e.g., ML {propto} MS3/4), Y represents a new observed scaling relation (e.g., ML {propto} DS2.0), and Z is the allometric conjunction of X and Y (e.g., MS {propto} DS8/3). The model was rejected whenever a predicted exponent for a particular Z failed to fall within the 95% confidence intervals of the corresponding observed exponent (see Table 3).


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  		Table 3. Matrix used to evaluate the "canonical" biomass partitioning pattern. Assertions of the model ("if X") and observed scaling relations ("and Y") conjoin ("then Z") and obtain a scaling exponent for leaf, stem, and root biomass (ML, MS, and MR, respectively) vs. basal stem diameter or plant height (DS and h, respectively). Observed exponents ({alpha}RMA and 95% confidence intervals) for Z are from Table 2. Accepted and rejected Z are indicated by + and –, respectively, in the last column

 

   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
LITERATURE CITED
 
Across species examined, leaf, stem, and root biomass scaled roughly as the 2-, 5/2-, and 9/4-power of stem diameter (Table 2). Analyses of residuals showed that these scaling relations were log-log linear; no correlation was observed for residual values and ecological factors such as latitude, plant density, or the average age of individuals within communities. However, inspection of bivariate plots revealed that the comparatively few outliers were data from conifer-dominated communities or managed conspecific stands (Fig. 1A–C).



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  		Fig. 1. Log-log bivariant plots of leaf, stem, and root biomass vs. basal stem diameter (ML, MS, and MR vs. DS). Data are redacted from the Cannell (1982) compendium. Original units are kilograms for organ biomass and meters for stem diameter. Solid lines are ordinary least squares regression curves for angiosperm and conifer data sets (see insert); curves are distinguished by "a" and "c" for angiosperms and conifers, respectively. Regression parameters are given in Table 2

 
Based on the 95% confidence intervals of these scaling exponents, the scaling of angiosperm and conifer organ biomass vs. stem diameter did not statistically differ from the scaling observed across all species. For example, across species, the 95% confidence intervals of the exponent for leaf biomass vs. stem diameter were 1.91–2.08, whereas those for angiosperm and conifer leaf biomass vs. diameter were 1.76–1.96 and 1.85–2.02, respectively (Table 2). For all comparisons, the scaling relation between stem biomass vs. diameter was the strongest statistically, whereas the relation for root biomass vs. stem diameter was the weakest.

Log-log linear relations were similarly observed for all organ biomass vs. height comparisons, although the data scatter for these plots was broader compared to that of biomass vs. diameter comparisons (Fig. 2A–C). Across all species, leaf and stem (and root) biomass scaled roughly as the 9/4- and 3-power of height, respectively (Table 2). The scaling exponents observed for the angiosperm and conifer data sets did not differ statistically from those observed for all species. However, the scaling of stem biomass vs. plant height for the angiosperm and conifer data sets differed based on the 95% confidence intervals of their exponents (Table 2).



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  		Fig. 2. Log-log bivariant plots of leaf, stem, and root biomass vs. plant height (ML, MS, and MR vs. h). Data are redacted from the Cannell (1982) compendium. Original units are kilograms for organ biomass and meters for height. Solid lines are ordinary least squares regression curves for angiosperm and conifer data sets (see insert); curves are distinguished by "a" and "c" for angiosperms and conifers, respectively. Regression parameters are given in Table 2

 
Across all species (and within the angiosperm and conifer data sets), plant height scaled roughly as the 8/9-power of stem diameter (Table 2). This near isometric relation was "internally consistent" with respect to the exponents observed for biomass vs. stem diameter or plant height. For example, across species, leaf biomass scaled as the 1.99-power of stem diameter and as the 2.24-power of height, i.e., ML {propto} DS1.99 {propto} h2.24. Thus, h {propto} DS1.99/2.24=0.89, whereas h {propto} DS0.89 was observed (Table 3). Likewise, across species, stem biomass scaled as the 2.55-power of diameter and as the 2.84-power of height. Accordingly, h {propto} DS2.55/2.84=0.90, whereas h {propto} DS0.89 was observed (Table 3).

In passing, the near log-log linear isometric relation for plant height vs. stem diameter is consistent with prior analyses indicating a geometric self-similarity between these two measures of plant size within the size ranges represented by the Cannell (1982) database, i.e., 0.20 m ≤ h ≤ 49.9 m (mean ± 1 SE = 13.4 ± 0.29 m) and 0.003 m ≤ DS ≤ 0.645 m (mean ± 1 SE = 0.16 ± 0.004) (Niklas, 1994b ). However, over a broader range of size and species phyletic affiliation (including non-vascular embryophytes), a log-log nonlinear relation between plant height and stem diameter is evident (Niklas, 1994a ).

The observed scaling relations collectively provide reasonable but not unequivocal support for the assertions that ML {propto} MS3/4 {propto} MR3/4 and MS {propto} MR hold across seed plant species with self-supporting stems (Table 3). For example, the scaling exponents predicted by the model for leaf, stem, and root biomass vs. plant height were 2.13, 2.95, and 2.84, respectively, whereas the 95% confidence intervals of observed scaling exponents were 2.07–2.41, 2.68–2.99, and 2.58–3.04, respectively. Likewise, based on the scaling of stem biomass vs. diameter, leaf biomass is predicted to scale as the 1.99-power of diameter, which is the observed exponent (Table 3). Similar results were obtained for the angiosperm and conifer data sets. The most glaring discrepancies between predicted and observed scaling relations are those related to root biomass.

Two factors influence whether the scaling relations collectively support or refute the "canonical" partitioning model (Niklas and Enquist, 2001 , 2002a , b ; Enquist and Niklas, 2002 ). First, the published reports compiled by Cannell (1982) probably underestimate fine and small root biomass. Second, most statistical protocols obtain highly conservative 95% confidence intervals for the scaling exponents (slopes) of regression curves involving two biologically interdependent variables. Underestimates of fine and small root biomass will necessarily reduce the numerical values of observed exponents when root biomass serves as the ordinate variable in regression analyses. A systematic and size-dependent underestimation of root biomass may explain why the scaling exponent observed for root biomass vs. stem diameter fails to agree with the expectation that root biomass scales isometrically with respect to stem biomass. By the same token, whereas narrow 95% confidence intervals for scaling exponents are highly desirable in allometric analyses, artificially conservative intervals will bias toward Type I error when observed and predicted scaling exponents are numerically compared. Thus, stringent protocols for calculating confidence intervals will favor rejecting valid hypotheses (Type I error).

The supposition that most reports systematically underestimate root biomass for increasingly larger plants is based on the contention that fine and small roots become increasingly more difficult to excavate for progressively larger or deeper root systems and on the empirical observation that fine and small roots tend to disproportionately increase with increasing plant size (e.g., Powell and Day, 1991 ; Makkonen and Helmissar, 2001 ). Across species or conspecifics differing in size, the resulting size-dependent bias will lower the numerical values of root biomass scaling exponents.

This "artifact" bears significantly on the observation that the scaling exponents for root biomass vs. stem diameter and stem biomass vs. stem diameter fail to numerically agree. For example, across species, root biomass and stem biomass scale as the 2.22- and 2.55-power of stem diameter. These exponents are numerically inconsistent with an isometric relation for root vs. stem biomass as judged by the scaling of organ biomass with respect to stem diameter. However, this is only true if 2.22 reflects the scaling of total root biomass with respect to stem diameter. In this respect, it is noteworthy that the scaling relations observed for conifer species (with presumably more shallow and easily excavated root systems than dicot trees) comply with an isometric relation between root vs. stem biomass.

Protocols for calculating the 95% confidence intervals of scaling exponents are of broad concern in allometric analyses. No generally accepted procedure currently exists, most probably underestimate confidence intervals, and thus most favor the rejection of true allometric hypotheses (see Sokal and Rohlf, 1981 ; Harvey and Mace, 1982 ; Peters, 1983 ; Rayner, 1985 ; McArdle, 1988 ; Niklas, 1994a ). The protocol used here was intentionally selected to provide conservative estimates of the confidence intervals of exponents, thereby favoring the rejection of the model's predictions.

It cannot be overly emphasized that the data sets used to evaluate the "canonical" model pertain to species with self-supporting stems and therefore are not representative of the full spectrum of seed plant growth habits. The model evaluated here assumes that leaves are the sole or principal photosynthetic organs. This assumption is frequently violated by species lacking woody stems (e.g., species with a herbaceous, rhizomatous, or liana growth habit) such that the model is invalid. Additional evaluations of the biomass partitioning model using larger or different databases reflective of nonarborescent growth habits are clearly required. However, based on the analyses presented here and the assertion that scaling exponents for root biomass allometric trends are "deflated" due to measurement error and thus suspect, the model obtains internally consistent predictions that comply reasonably well with the scaling exponents reported here for the Cannell (1982) compendium.


   FOOTNOTES
 
1 The author thanks Prof. William E. Stein, Jr. (Department of Biological Science, SUNY–Binghamton, NY), who as an Associate Editor of the American Journal of Botany supervised the review process and served as the acting Editor-in-Chief for this manuscript; Kenneth C. Cheung (Cornell) for research assistance; two anonymous reviewers who offered insightful and constructive comments; and the New York State Hatch granting agency and the Alexander von Humboldt Stiftung for support. Back

2 kjn2{at}cornell.edu ; phone: (607) 255-8727; FAX: (607) 255-5407 Back


   LITERATURE CITED
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 LITERATURE CITED
 
Cannell M. G. R. 1982 World forest biomass and primary production data. Academic Press, London, UK

Enquist B. J. K. J. Niklas 2002 Global allocation rules for patterns of biomass partitioning in seed plants. Science 295: 1517-1520[Abstract/Free Full Text]

Harvey P. H. G. M. Mace 1982 Comparisons between taxa and adaptive trends: problems of methodology. In King's College Sociobiology Group [eds.], Current problems in sociobiology, 343–361. Cambridge University Press, Cambridge, UK

Makkonen K. H. S. Helmissar 2001 Fine root biomass and production in Scots pine stands relative to stand age. Tree Physiology 21: 193-198[ISI][Medline]

McArdle B. H. 1988 The structural relationship: regression in biology. Canadian Journal of Zoology 66: 2329-2339

Niklas K. J. 1994a Plant allometry. University of Chicago Press, Chicago, Illinois, USA

Niklas K. J. 1994b The scaling of plant and animal body mass, length, and diameter. Evolution 48: 44-54[CrossRef][ISI]

Niklas K. J. B. J. Enquist 2001 Invariant scaling relationships for interspecific plant biomass production rates and body size. Proceedings of the National Academy of Sciences, USA 98: 2922-2927[Abstract/Free Full Text]

Niklas K. J. B. J. Enquist 2002a On the vegetative biomass partitioning of seed plant leaves, stems, and roots. American Naturalist 159: 482-497[CrossRef][ISI]

Niklas K. J. B. J. Enquist 2002b Canonical rules for plant organ biomass partitioning and growth allocation. American Journal of Botany 89: 812-819[Abstract/Free Full Text]

Peters R. H. 1983 The ecological implications of body size. Cambridge University Press, Cambridge, UK

Powell S. W. F. P. Day Jr 1991 Root production in four communities in the Great Dismal Swamp. American Journal of Botany 78: 288-297[CrossRef][ISI]

Rayner J. M. V. 1985 Linear relations in biomechanics: the statistics of scaling functions. Journal of Zoology, London A206: 415-439

Sokal R. R. F. J. Rohlf 1981 Biometry. W. H. Freeman, New York, New York, USA




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