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Ecology |
2Department of Plant Biology, Cornell University, Ithaca, New York 14853-5908 USA; 3Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721 USA
Received for publication September 25, 2001. Accepted for publication November 30, 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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MS3/4 MR3/4 such that MS MR and that GL GS GR. A large synoptic data set for standing plant organ biomass and organ biomass production spanning ten orders of magnitude in total plant body mass supports these predictions. Although the numerical values for the allometric "constants" governing these scaling relationships differ between angiosperms and conifers, across all species, standing leaf, stem, and root biomass, respectively, comprise 8%, 67%, and 25% of total plant biomass, whereas annual leaf, stem, and root biomass growth represent 30%, 57%, and 13% of total plant growth. Importantly, our analyses of large data sets confirm the existence of scaling exponents predicted by theory. These scaling "rules" emerge from simple biophysical mechanisms that hold across a remarkably broad spectrum of ecologically and phyletically divergent herbaceous and tree-sized monocot, dicot, and conifer species. As such, they are likely to extend into evolutionary history when tracheophytes with the stereotypical "leaf," "stem," and "root" body plan first appeared.
Key Words: allometry • biomass allocation • organ biomass • plant growth
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Hunter and Lloyd, 1987
; Charnov, 1993
; Iwasa, 2000
). This paper attempts to address this issue, first, by reviewing an allometric model predicting the scaling exponents for vegetative organ biomass partitioning and growth, and, second, by testing whether the scaling exponents predicted by this theory comply with those observed for newly gathered data sets for plant organ biomass partitioning and annual allocation.
The model reviewed here rests on the general allometric equation log Y1 = log ß + 
log Y2, where Y1 and Y2 are interdependent (size or growth) variables, ß is the allometric constant, and is the scaling exponent (Huxley, 1932
; Gould, 1966
; Peters, 1983
; LaBarbera, 1986
; Niklas, 1994
; Enquist and Niklas, 2001
, 2002; Niklas and Enquist, 2001
, 2002). Previous theoretical and empirical studies using this formula have shown that, across 20 orders of magnitude of body size (ranging from unicellular algae to that of mature trees), total plant growth GT (biomass production per plant per year) is proportional to the 3/4-power of body biomass MT and that total plant growth scales isometrically with respect the capacity of an individual to capture sunlight H (i.e., GT M3/4 and GT H, respectively; Enriquez et al., 1996
; Niklas and Enquist, 2001
). Other recent developments in allometric theory have shed light on a broad spectrum of other ecological phenomena ranging from community dynamics to vascular plant hydraulics and other life-history traits (West, Brown, and Enquist, 1997, 1999
; Enquist et al., 1999
; Enquist and Niklas, 2001
, 2002; Niklas and Enquist, 2002).
Here, with the aid of a few simple biophysical assumptions, we further elaborate on allometric theory that predicts standing leaf biomass will scale as the 3/4-power of stem (or root) biomass and that stem and root biomass will scale isometrically with respect to one another. We also review the model that predicts annual leaf, stem, and root biomass production (= organ growth) will each scale isometrically with respect to one another.
To test these predictions, we present statistical analyses of a large data base for organ biomass and growth representing a broad spectrum of seed plant species, which has been recently expanded to cover ten orders of magnitude in total body size. Using these data, we show that the scaling exponents predicted by our model comply statistically with those observed empirically. Although substantial variation in the allometric constants for these scaling relationships exists as a result of species-specific differences in ontogeny, anatomy, habitat preferences, and other important phenotypic and environmental features, the numerical values of empirically determined allometric constants can be used to calculate the standing organ biomass and the annual organ growth for an "average" plant across all species as well as an "average" angiosperm or conifer species.
The model and analyses presented here identify a single canonical set of rules for the proportional relationships among organ biomass partitioning and allocation across a remarkably diverse spectrum of vascular land plant species. These rules provide a potentially powerful analytical tool for ecological and evolutionary studies, especially since they likely extend into the fossil record when the stereotypical tracheophyte "leaf," "stem," and "root" body plan first evolved.
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ALLOMETRIC THEORY |
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TOPABSTRACTINTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Standing organ biomass relationships
The scaling exponents governing the relationships among standing leaf, stem, and root biomass (ML, MS, and MR, respectively) are derived analytically based on the theoretical prediction and empirical observation that total annual plant growth GT scales as the 3/4-power of total plant biomass MT (which equals the sum of standing leaf, stem, and root biomass; ML, MS, and MR, respectively) and that it also scales isometrically with respect to ML (Niklas and Enquist, 2001
; see also Enriquez et al., 1996
). Thus, GT = ß1 MT3/4 = ß1 (ML + MS + MR)3/4 = ß2 ML, where the allometric constants ß1 and ß2 have units including per year. Simplifying this relationship gives ML = ß3 (ML + MS + MR)3/4, where ß3 = ß1/ß2. Since MS = ß4 
S DS2 LS and MR = ß5 R DR2 LR, where is tissue bulk density, L is total organ length, and D is diameter, and since ß4 S and ß5 R are roughly constant for each species (i.e., ß4 S = ß6 and ß5 R = ß7, respectively), we obtain ML = ß3 [ML + ß6 DS2 LS + ß7 DR2 LR]3/4.
This last relationship can be solved for ML because the total volume of water absorbed and transported by roots through stems to leaves per unit time must be conserved such that DS2 DR2 and because, for metabolic as well as hydraulic reasons, ML is predicted to be proportional to total stem and root diameter such that ML = ß8DS2 = ß9DR2, where ß8 and ß8 are additional allometric constants (see Murray, 1927
; Kramer, 1983
; Zimmerman, 1983
). Therefore, ML = ß34 [1 + (ß6/ß8) LS + (ß7/ß9) LR]3. Provided that root and stem length scale isometrically such that LR = ß10 LS, then ML = ß34 [(1/LS) + (ß6/ß8) + (ß7 ß10/ß9)]3 LS3. Noting that 1/LS 
0 with increasing growth in size, it is clear that ML 
ß34 [(ß6/ß8) + (ß7 ß10/ß9)]3 LS3 = ß11 LS2.
Allometric theory thus obtains MS = (ß6/ß8ß111/3) ML ML1/3 = ß12 ML4/3, MR = (ß7 ß10/ß9 ß111/3) ML ML1/3 = ß13 ML4/3 and, MS = (ß12/ß13) MR = ß14 MR, or, in terms of simple proportional terms, ML MS3/4, ML MR3/4 and MS MR. Also, the relationship between standing shoot and root biomass is predicted to be ML + MS = (ß12/ß13) MR + (MR/ß13)3/4. Numerical simulations using different values for ß12 and ß13 indicate that ML + MS will, on average, scale in a near isometric way with respect to MR. Importantly, these scaling relationships are expected to be invariant with respect to species phyletic affiliation provided that leaves are the sole or principal photosynthetic organs.
Annual organ growth relationships
The scaling exponents for leaf, stem, and root annual biomass production rates or annual "organ growth" (GL, GS, and GR, respectively) are derived analytically by noting that total annual plant growth GT must equal the sum of annual leaf, stem, and root growth and that total growth scales isometrically with respect to standing leaf biomass. Thus, GT = GL + GS + GR = ß1 ML, where the allometric constant ß1 includes units of per year (see Niklas and Enquist, 2001
, 2002).
For deciduous species, annual leaf biomass production is directly proportional to standing leaf biomass such that GL = ß2 ML (Enriquez et al., 1996
; Niklas and Enquist, 2001
). Therefore, GT = (ß1/ß2) GL and thus GL = [ß2/(ß1 – ß2)] (GS + GR) = ß3 GN, where GN denotes annual non-photosynthetic biomass production. However, for nondeciduous species, leaf growth is proportional to the difference between the standing leaf biomass and the leaf biomass retained from previous growth seasons, denoted here as Ml. That is, GL = ß4 (ML – Ml). For metabolic and life-history reasons (i.e., leaf phenology; see Ackerly and Reich, 1999
), we assume that the leaf biomass retained from previous seasons is directly proportional to the standing leaf biomass in any growth season. If so, then Ml = ß5 ML from which it follows that GL = ß4 (1 – ß5) ML = ß6 ML and GT = (ß1/ß6) GL = ß7 GL such that GL = (ß7 – 1)–1 (GS + GR) = ß8 GN. Thus, regardless of leaf phenology, an isometric relationship is predicted for annual leaf biomass production and the sum of annual stem and root biomass production. That is, GL = ß9 (GS + GR), where ß9 = ß3 or ß8.
This last scaling relationship may now be used to derive the scaling exponents relating leaf growth to stem and root growth for all species as follows. Stem (or root) growth in biomass must equal the product of some allometric constant ß, the bulk density of newly formed organ tissues , and organ volume, which for stems or roots equals D2L. Therefore, GS = ß10 S DS2 LS and GR = ß11 R DR2 LR. For any species, the allometric constants and bulk tissue densities relating to the construction of new stem or root tissues are relatively constant such that ß10 S = ß12 (and ß11 R = ß13). Thus, GS = ß12 DS2 LS and GR = ß13 DR2 LR such that GL = ß9 (ß12 DS2 LS + ß13 DR2 LR).
Since the water mass from roots to stems must be conserved, root and stem cross-sectional areas are expected to scale isometrically such that DR2 = ß14 DS2 (Niklas, 1994
). Assuming that annual root and stem extension in length is relatively constant for any particular species such that LR/LS = ß15, we see that GL = ß9 (ß12 + ß13 ß14 ß15) DS2 LS = ß16 DS2 LS. Since DS2 LS is proportional to GS, leaf and stem growth will scale isometrically with respect to one another: GL = (ß16/ß12) GS = ß17GS. Leaf growth will also scale isometrically with respect to root growth: GL = [ß9 ß17/(ß17 – ß9)] GR = ß18 GR. Thus, stem and root growth are predicted to scale isometrically: GS = [ß9/(ß18 – ß9)] GR = ß19 GR. Finally, annual shoot biomass production (i.e., the sum of leaf and stem growth, GL + GS) is predicted to scale isometrically with respect to root growth: GL + GS = ß17 GS + ß18 GR = (ß17 ß19 + ß18) GR = ß20 GR. Thus, in simple proportional terms, our model predicts GL GS, GL GR, GS GR, and GL + GS GR. These exponents are expected to be invariant with respect to species phyletic affiliation provided that leaves are the sole or principal photosynthetic organs.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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who compiled data sets for tree-sized dicot, monocot, and conifer species as well as a limited number of bamboo species.
Each of the Cannell data sets is standardized to 1.0 ha and represents 600 sites worldwide, published in a standardized tabular format that provides the primary citation and, when supplied by authors, longitude, elevation, the age of the dominant species (or conspecific in the case of monotypic managed stands), the number of plants per 1.0 ha ("plant density"), height, total basal stem cross-sectional area, and the standing biomass and net biomass production of stem wood, bark, branches, fruits, foliage, and roots (in units of metric tons of dry matter per year). The reported values for annual stem wood, bark, foliage, etc., production reflect as much as possible annual losses of dry matter due to mortality, litter-fall, decay, and consumption (see Cannell, 1982
).
Organ biomass and productivity were determined by authors from direct measurements of fully dissected representative plants (typically 
5 individuals) for the majority of the Cannell sites. Authors regressed these data to estimate total organ biomass per 1.0 ha community sample. Data based on estimated regression variables were rejected when entering the Cannell data sets into computer memory. Importantly, most of these data sets are for even-aged conspecific stands (n = 600 out of 880 usable data sets), and biomass production values are typically averaged values for two or more years. Therefore, for each site used in our analyses, the variance in standing organ biomass and biomass production was assumed to be comparatively small and annual production rates were considered representative of "normal" rather than idiosyncratic growth seasons.
Standing leaf, stem, and root biomass per "average" plant was computed for each of the Cannell sites using the quotient of total community standing organ biomass and plant density. Annual leaf, stem, and root production rates were similarly calculated using the quotient of annual organ type production per hectare sample and plant density. However, we note that most of the Cannell data sets probably underestimate standing root biomass and biomass production, particularly those of fine and small roots, because these are more difficult to excavate completely for increasingly larger root systems and because these root-size categories are reported to increase with increasing plant size (e.g., Powell and Day, 1991
). Thus, numerically higher scaling exponents than those predicted were anticipated for any regression analysis using root biomass or biomass production as the Y2 variable.
Since the Cannell data sets emphasize mature and large plant body sizes, additional data were recently gathered by K. J. Niklas from the primary literature published between 1990 and 2001 (data available on request) for species with comparatively small mature body sizes (e.g., Arabidopsis, Bromus, Lactuca, Lycopersicum, Plantago, Spartina) or for seedlings and saplings of tree species (e.g., Betula, Quercus, and Thuja). These additional data, which span 51 species not represented in the Cannell data sets, are from laboratory or field studies of plants grown under normal field or experimental conditions (e.g., elevated CO2, UV-B radiation, salinity, or soil micronutrient levels). Every attempt was made to select data evincing little variance per treatment as gauged by the standard errors reported for standing biomass or biomass production. For these plants, standing organ biomass and annual organ biomass production were calculated as for the Cannell data sets.
Statistical analyses
Model Type II (reduced major axis, RMA) regression analysis was used to determine scaling exponents (RMA) and allometric constants (ßRMA). The values for RMA and ßRMA were computed using the formulas RMA = OLS/r and log ßRMA = log 
1 – RMA log 2, where OLS is the ordinary least squares (OLS) regression exponent, r is the OLS correlation coefficient, and denotes the mean value of Y1 or Y2 (Sokal and Rohlf, 1981
; Niklas, 1994
). 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 (Sokal and Rohlf, 1981
; Harvey and Mace, 1982
; Rayner, 1985
; McArdle, 1988
). The numerical values of RMA and ßRMA differ little from those obtained from Model Type I (OLS regression) analyses whenever r2 
0.95 (Sokal and Rohlf, 1981
; Niklas, 1994
). Since r2 > 0.95 was found for most of the empirically determined interspecific relationships, the selection of regression model is arguably moot.
Paired comparisons of the untransformed data evinced linear trends when plotted on log-log scales (based on analyses of residuals and smoothing spline-regression models with different 
values). However, all of the values computed for the Cannell data sets come from populations differing in plant density and, given the nature of these data sets, the variance about the "mean" values for each growth variable could not be determined, yet undoubtedly differed across community sites. To reduce the resulting effects of heteroscedasticity, the raw data were log10-transformed for subsequent Model Type II regression analyses. This protocol is recommended for functional analyses of biological growth variables (Sokal and Rohlf, 1981
). Attempts to approximate trends in the log-transformed data with log-curvilinear regression models either failed to improve or reduced the goodness of fit (based on analyses of residuals, bivariate normal ellipse protocol estimates, or the correlation coefficients of spline-smoothing regression curves).
Regression analyses were performed on the pooled data sets to determine interspecific trends, on the angiosperm and conifer data sets separately to determine the effect of phyletic affiliation on regression parameters, and on individual species for which data were sufficient to determine intraspecific scaling relationships. Also, since the magnitude of many variables correlate with total standing biomass, paired values for organ biomass and growth rates were regressed over different ranges of their magnitude to determine the effect of plant size on the numerical values of scaling exponents. Since some primary sources did not report all the data needed to calculate values for all variables of interest, sample sizes n varied across statistical comparisons. The effect of n on regression parameters was determined using analyses of residuals.
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RESULTS |
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TOPABSTRACTINTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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DS2.
Another critical assumption was that annual leaf biomass production scales as the 2-power of stem diameter. Across all species, GL scaled as the 1.89 (±0.041)-power of DS (95% CI = 1.80–1.98; n = 410, r2 = 0.809, F = 1728, P < 0.0001). For angiosperms, GL scaled as the 1.79 ± 0.053 power of DS (95% CI = 1.67–1.90; n = 172, r2 = 0.852, F = 977.9, P < 0.0001). For conifers, the scaling exponent was 1.97 ± 0.058 (95% CI = 1.85–2.10; n = 235, r2 = 0.802, F = 941.5, P < 0.0001). Therefore, our data were consistent with the assumption that GL DS2.
The scaling exponents observed for standing biomass relationships did not significantly differ from those predicted by theory and bivariate plots had remarkably few statistical outliers (Fig. 1). Based on analyses of regression residuals and the 95% confidence intervals of log-log linear regression curves, standing leaf biomass scaled, on average, as the 3/4-power of stem biomass, both across all species and within the angiosperm and conifer data sets (Table 1). Likewise, stem and root biomass scaled isometrically with respect to each other.
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1–2 yr old) whose "standing" organ biomass is equivalent to annual organ biomass production (e.g., ML GL). For these plants, allometric theory predicts a scaling exponent of 1.0 rather than 0.75. This was demonstrated by regression and statistical comparisons between leaf biomass vs. stem (or root) biomass within the small and the large size ranges of data collected from juvenile and older plants (i.e., 10–6 < MS(or R) 10–3 and 10–3 < MS(or R) 10+4 kg dry mass/plant, respectively). Within the small stem (or root) biomass range (data from juveniles), leaf biomass scaled isometrically, as predicted when "standing" biomass equals annual organ growth. In contrast, leaf biomass scaled as the 3/4-power within the large size range of stem or root biomass (data from mature individuals), as predicted by theory (Table 2).
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; Makkonen and Helmissar, 2001
). These features are expected to elevate the slope of any regression curve with root biomass production plotted on the abscissa, which was evident throughout all statistical comparisons (see Tables 1–3).
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25) were limited in number, this prediction was consistent with the allometry of those species for which sufficient sample sizes existed (Fig. 4). For example, for the conifer Cryptomeria japonica, the standing leaf biomass scaled as the 0.74 (±0.031)-power of stem biomass (95% CI = 0.64–0.77; n = 59, r2 = 0.854, F = 335.0, P < 0.0001) and as the 0.72 (±0.041)-power of root biomass (95% CI = 0.64–0.81; n = 30, r2 = 0.913, F = 292.7, P < 0.0001). Standing stem biomass also scaled as the 0.99 (±0.016)-power of root biomass (95% CI = 0.93–1.0; n = 30, r2 = 0.992, F = 3543, P < 0.0001). The exponents for these scaling relationships were statistically indistinguishable from those predicted by theory (i.e., 3;cl4, 3;cl4, and 1, respectively). Also, for C. japonica, annual leaf biomass production scaled as the 1.08 (±0.05)-power of annual stem biomass production (95% CI = 0.963–1.19, r2 = 0.828, n = 71, F = 332.5, P < 0.0001) and as the 1.06 (±0.04)-power of annual root biomass production (95% CI = 0.964–1.15, r2 = 0.890, n = 66, F = 519.3, P < 0.0001). Stem biomass production scaled as the 0.99 (±0.03)-power of root biomass production (95% CI = 0.920–1.05, r2 = 0.936, n = 66, F = 935.8, P < 0.0001). These scaling exponents did not differ from 1.0, as predicted by our model.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONALLOMETRIC THEORYMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Bazzaz and Grace, 1997
; Cornelissen, 1999
; Poorter and Nagel, 2000
). However, the dimensionless organ growth quotients GL/GS and GL/GR, which equal the respective allometric "constants" for these scaling relationships, can be used to calculate empirically the percentage of leaf, stem, and root growth with respect to total annual growth. For example, across all species, the quotients GL/GS and GL/GR equal 0.53 and 2.27, respectively (Table 4). Thus, for each 1.0 kg of leaf biomass produced per year per plant (i.e., GL = 1.0 kg/yr), there are (1.0 kg/yr)/0.53 = 1.89 kg/yr and (1.0 kg/yr)/2.27 = 0.44 kg/yr of stem and root growth in biomass, respectively. Since there is a total of 3.33 kg/yr (=1.0 kg/yr + 1.89 kg/yr + 0.44 kg/yr) annual biomass production, the percentages of leaf, stem, or root annual growth are 30% ({[1.0 kg/yr]/3.33 kg/yr} x 100%), 57% ({[1.89 kg/yr]/3.33 kg/yr} x 100%), and 13% ({[0.44 kg/yr]/3.33 kg/yr} x 100%), respectively (Fig. 5A). Using the same procedure, annual angiosperm leaf, stem, and root biomass growth respectively equals 32%, 59%, and 9% of total plant growth, whereas for conifers, the respective values are 48%, 35%, and 17%. On average, conifers appear to "invest" slightly less annual growth in the construction of new root tissues compared to the majority of angiosperm species.
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). That angiosperm stem and root biomass, on average, slightly exceeds that of conifers resonates with the higher densities of angiosperm woods with respect to those of most conifer woods (Tendelenburg and Mayer-Wegelin, 1955
; Zobel and Buijenen, 1989
; Niklas, 1994
). At the level of individual species, the differences in the numerical values of allometric constants account for much of the "data spread." This is a direct reflection of species-specific differences in anatomy, morphology, and ontogeny (see Niklas and Enquist, in press). Nonetheless, the regression curves for individual species share the same slopes (scaling exponents). When seen from this broad interspecific perspective, a single invariant pattern for organ biomass partitioning and annual biomass allocation to organ construction is identified by both our model and statistical analyses of large data sets. Remarkably, this pattern holds across a minimum of ten orders of magnitude of total body size and at least eight orders of magnitude in organ growth rates. It is indifferent to species phyletic affiliation, since angiosperms and conifers manifest the same scaling exponents, and it is largely insensitive to local environmental conditions, since data were gathered from plants growing under normal as well as some strikingly stressful conditions (e.g., low light levels, elevated UV-B, varying soil salinity, impoverished soil nutrients, or drought).
This invariance suggests that the partitioning of a finite amount of biomass produced per year (among three functionally equally important organ types) necessitates tradeoffs that are governed at least in part by the inexorable operation of biophysical phenomena (e.g., the conservation of water mass flowing from roots through stems to leaves and the mechanical relationship between leaf biomass and the stresses it produces in subtending stems). Whether these tradeoffs are resolved optimally over the course of an individual's ontogeny or over the evolutionary history of each species remains conjectural (see Hunter and Lloyd, 1987
; Schieving, 1988
; Poorter, 1989
; Lloyd and Venable, 1992
; Iwasa, 2000
; Niklas and Enquist, 2002). Yet, it is clear that otherwise widely different seed plant species are convergent in terms of the allometry of biomass partitioning and allocation.
This convergence provides a potentially powerful analytical tool for macroecological and evolutionary analyses of plant communities. For example, our data and model indicate that root biomass can be predicted with reasonable accuracy based on aboveground biomass measurements. This has obvious implications to modeling global climate change affected by standing plant biomass (see Raich and Nadelhoffer, 1989
). Likewise, there is good reason to suspect that the pattern of biomass partitioning and allocation extends throughout the evolutionary history of vascular plants possessing a stereotypical "leaf," "stem," and "root" body plan (Niklas, 1997
). If so, then root biomass may be estimated provided that aboveground biomass can be measured or reasonably estimated from morphological data.
Additional theoretical and empirical insights are required, especially in terms of reproductive biomass allocation, which can vary from year to year. But it is becoming increasingly clear that allometric theory is rapidly developing and holds much promise to shed considerable light on virtually every aspect of plant life, past and present.
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FOOTNOTES |
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4 Author for reprint requests (phone: 607-255-8727; Fax: 607-255-5407; kjn{at}2cornell.edu
)
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  K. J. Niklas and H.-C. Spatz Allometric theory and the mechanical stability of large trees: proof and conjecture Am. J. Botany, June 1, 2006; 93(6): 824 - 828. [Abstract] [Full Text] [PDF]
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 S. SUN, D. JIN, and P. SHI The Leaf Size-Twig Size Spectrum of Temperate Woody Species Along an Altitudinal Gradient: An Invariant Allometric Scaling Relationship Ann. Bot., January 1, 2006; 97(1): 97 - 107. [Abstract] [Full Text] [PDF]
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K. J. NIKLAS Modelling Below- and Above-ground Biomass for Non-woody and Woody Plants Ann. Bot., January 2, 2005; 95(2): 315 - 321. [Abstract] [Full Text] [PDF]
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K. J. Niklas Reexamination of a canonical model for plant organ biomass partitioning Am. J. Botany, February 1, 2003; 90(2): 250 - 254. [Abstract] [Full Text] [PDF]
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