The balanced-growth hypothesis and the allometry of leaf and root biomass allocation


  • B. Shipley,

    Corresponding author
    1. Département de biologie, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada, and
      †Author to whom correspondence should be addressed. E-mail:
    Search for more papers by this author
  • D. Meziane

    1. Département de biologie, Université Sidi Mohamed Ben Abdellah, Fès, Morocco
    Search for more papers by this author

†Author to whom correspondence should be addressed. E-mail:


1. Many ecological models of plant growth assume balanced growth: that biomass is allocated preferentially to leaves or roots to increase capture of the limiting external resource. An alternative explanation is based on nonlinear (allometric) allocation as a function of plant size. The objective of this study was to test between these two alternative explanations.

2. A total of 1150 plants from 22 different herbaceous species were grown in hydroponic sand culture in factorial combinations of high (1100 µmol m−2 s−1) and low (200 µmol m−2 s−1 PAR) irradiance crossed with a full-strength and a 1/6 dilution of Hoagland’s hydroponic solution. Plants were harvested at 15, 20, 25, 30 and 35 days postgermination, and dry mass was determined for leaf and root components. These data were used to test the hypotheses of balanced growth and of allometric allocation.

3. Both irradiance and nutrient supply affected the slope and intercept of the root : shoot allometry, contrary to the allometric hypothesis but in agreement with the hypothesis of balanced growth; decreased nutrient supply increased allocation to roots; and decreased irradiance increased allocation to leaves.

4. Plants allocated relatively more biomass to roots than to leaves as plants grew larger. In order for the balanced-growth hypothesis to be correct, the net rate of nutrient uptake per unit root mass must have been decreasing relative to the net rate of carbon gain per unit leaf mass.

5. We suggest two reasons why this might be the case: (i) older roots decreased their efficiency of nutrient uptake; and (ii) larger root systems more rapidly decreased the available nutrients between flushes of hydroponic solution.

6. These results support the notion of balanced growth that is found in many ecological models of plant growth.


As a plant captures carbon and nutrients, it must allocate these acquired resources to new tissues. Allocation of these newly acquired resources to different tissues or plant parts will then affect the subsequent rates of capture of carbon and soil resources. Differential biomass allocation therefore has profound implications for plant growth. Hypotheses concerning different ‘strategies’ of biomass allocation are at the heart of many theories in plant ecology and evolution (Aarssen & Taylor. 1992; Bazzaz et al. 1987; Grime 1977; Grime 1979; Hilbert 1990; Lovett Doust 1989; Perrin 1992; Thornley 1998; Tilman 1988; Tilman 1990; Westoby 1998). Although many models, of varying degrees of complexity, have been proposed to describe such allocation rules, this paper concentrates on two simple, alternative rules that have been advanced in the literature. The first hypothesized rule is that of balanced growth (Davidson 1969; Garnier 1991; Hunt 1975); the second is of allometric allocation (Müller, Schmid & Weiner 2000).

The balanced-growth hypothesis is incorporated into many different models. This hypothesis has generally been tested by computing ratios of root to leaf mass (or root to shoot ratios). However, if the allocation of new biomass to roots and leaves follows an allometric relationship whose slope is not unity, then such ratios will change as plants grow. As plants growing under nutrient or light limitation will generally be smaller at a given age than plants growing under more favourable conditions, changes in root : leaf ratios, when comparing similarly aged plants growing under different growth conditions, may simply reflect this underlying allometric principle. This explanation underlies the allometric allocation hypothesis of Müller et al. (2000) with respect to biomass allocation. The objective of this paper is to evaluate these two hypotheses.

Balanced-growth hypothesis

Intuitively, the notion of balanced growth is simply that the plant will preferentially allocate biomass to the plant organ that is harvesting the resource limiting growth. Because carbon is captured by leaves, while water and mineral nutrients are captured by roots, this means that biomass allocation will favour leaves if light becomes more limiting, and will favour roots if the mineral nutrient becomes limiting to growth. The hypothesis of balanced growth can be more formally derived from a simple biological argument. Each biological molecule has a specific stoichiometry. For instance, each molecule of chlorophyll a has 55 carbon atoms and four nitrogen atoms. If a plant were to allocate resources to leaves versus roots to maximize the production of chlorophyll a, then this must be done such that the net rate of carbon to nitrogen acquisition should be 55 : 4; any other allocation rule would result in an expenditure of energy and an accumulation of either carbon or nitrogen that could not be converted into chlorophyll a. Plants consist of many different molecules whose relative abundances change over ontogeny, but the general argument still holds. Where ML and MR are the dry mass of photosynthetic organs (leaves) and roots; Am and Um are the net whole-plant rates of carbon assimilation and nutrient uptake per unit leaf or root mass; and N and C are the mass of the limiting nutrient and carbon, the hypothesis of balanced growth can be equivalently formalized as:

ML · Am ∝ MR · Um    eqn 1a
image(eqn 1b)
image(eqn 1c)

Rearranging equation 1a and taking logarithms, we obtain:

ln(ML) ∝ ln(MR) + ln(Um) − ln(Am)    eqn 2a
ln(ML) = α + βln(MR) + δ1ln(Um) − δ2ln(Am)    eqn 2b

The partial intercept (α) and partial slope β quantify the allometric relationship between leaf and root mass, and the partial slopes δ1 and δ2 quantify how far changes in the net whole-plant rates of carbon assimilation and nutrient uptake per unit leaf or root mass change the overall intercept. Equation 2(b) is useful because it can be directly compared to the hypothesis of allometric allocation. The following predictions can be obtained. First, the partial slope β describing the root–leaf allometry should be independent of resource supply rates. Second, increasing the soil resource supply rates from one constant amount to another should increase the overall intercept by increasing the whole-plant nutrient uptake per unit root mass (MR). Third, increasing the irradiance supply rate from one constant amount to another should decrease the overall intercept by increasing the whole-plant rates of carbon assimilation per unit leaf mass (AM).

Hypothesis of allometric allocation

Müller et al. (2000) grew plants of 27 herbaceous species in a high- or low-nutrient environment, and fitted (their Table 5) allometric regression models of the form:

ln(MR) = a + bln(ML) + N + N · ln(ML) + ɛ    eqn 3

where N was a two-level factor indicating the experimental nutrient level experienced by the plant, and ɛ is the residual deviation. An increasing nutrient supply, as produced in the higher nutrient environment provided by Müller et al. (2000), would increase the rate of uptake of nutrients per unit root mass (Um). Comparing equations 3 and 2, and noting that the order of dependent and independent variables is reversed in these two equations, Müller et al. (2000) tested and rejected the balanced-growth hypothesis. There was no significant difference in either allometric slopes or intercepts when comparing plants grown in high- versus low-nutrient environments, as shown by nonsignificant N and N · ln(ML) terms in their model. Twenty-two of 27 species allocated higher proportions of new biomass to leaves than to roots as they grew, such that small plants had higher root : leaf ratios, that is, they had allometric leaf versus root slopes less than 1·0, and none had a slope significantly greater than 1·0. The root : leaf ratios did change as expected given the hypothesis of balanced growth, as plants in the nutrient-limited environment, being smaller, had higher root : leaf ratios. However, these changes were consistent with a simple allometric relationship that was not affected by changes in nutrient supply.

Based on such results, Müller et al. (2000) suggested that, rather than the plastic allocations in response to different resource availabilities of the balanced-growth hypothesis, allocation patterns are more parsimoniously explained as allometric strategies in which proportionally more biomass is allocated to leaves than to roots as plants grow. If this result is generally true, the balanced-growth hypothesis must be rejected. Such a rejection would put into question the many published models of biomass allocation based on the balanced-growth hypothesis.

To test between the balanced-growth and allometric allocation hypotheses, plants can be grown in environments differing in supply rates of light and mineral nutrients, with tests for significant changes in allometric slopes and intercepts when comparing across environments. This paper reports such a test.

Materials and methods

Experimental design

The experiment consisted of 1150 plants from 22 different species. (Species are listed in Table 3.) Each plant grew in a separate 1·3 dm3 container in washed silica sand in a growth chamber with a 15/9 h light : dark cycle, a day : night temperature of 25/15 °C, and a relative humidity above 80%. Each plant grew in one of four resource environments: high (L, 1100 µmol m−2 s−1) and low (l, 200 µmol m−2 s−1) irradiance crossed with high (N, full-strength Hoagland’s nutrient solution) and low (n, 1/6 dilution) external nutrient concentrations. These four experimental treatments are termed LN, Ln, lN and ln treatments, respectively. Each 1·3 dm3 container was filled to field capacity with the nutrient solution three times a day. Plants were harvested at 15, 20, 25, 30 and 35 days postgermination. In general, three plants per species per experimental treatment were harvested at each harvest date, but this number was reduced in a few cases. Eupatorium maculatum was not grown in the lN treatment, and Hordeum jubatum was not grown in the Ln treatment. More details of this experiment, including the nutrient solution, are given by Meziane & Shipley (1999).

Table 3.  Slopes (b) of allometric regressions of the form: ln(ML) = a + bln(MR) + ɛ fitted separately to each species in each treatment group; MR = dry mass of roots, ML = dry mass of leaf lamellas
  1. Values in bold indicate slopes significantly different (P < 0·05) from unity. Treatments: L = 1100 µmol m−2 s−1 PAR; l = 200 µmol m−2 s−1 PAR; N = full-strength hydroponic solution, n = 1/6 dilution.

Acorus calamus L.0·8650·9450·9320·847
Agropyron repens (L.) Beav.0·9151·1380·9160·789
Bromus inermis Leyss.0·8750·7590·8290·779
Carex crinita Lam.0·8650·7101·1330·933
Chrysanthemum leucanthemum L.1·0990·9751·0291·052
Cichorium intybus L.0·8840·7920·8150·719
Deschampsia cespitosa (l). Beav.1·0631·2611·1070·947
Erysimum cheirantoides L.0·8500·8510·8390·727
Eupatorium maculatum L.0·8520·559 0·885
Hieracium aurantiacum L.0·9480·9480·9170·923
Hordeum jubatum L.0·527 0·9890·858
Leontodon autumnalis L.0·9200·9141·0560·831
Oenothera biennis L.0·9750·6201·0490·939
Panicum capillare L.0·9520·8691·0000·944
Phleum pratense L.0·7380·9141·0770·980
Plantago lanceolata L.0·8820·8450·8920·830
Plantago major L.0·8660·7030·9480·981
Poa pratensis L.1·0400·8781·2360·806
Polygonum lapathifolium L.0·9100·9360·9061·036
Prunella vulgaris L.1·0031·0091·0560·940
Rumex acetosa L.0·7480·6710·7210·611
Silene cucubalus Wibel.0·9870·8310·8040·723

Each plant was separated into four parts: leaf lamina, stems + leaf petioles, rhizomes (when present), and reproductive tissues (in only a few plants at the end of the experiment). These biomass compartments were dried at 70 °C for a minimum of 36 h. This paper deals only with mass allocation to the leaf and root compartments.

Statistical analysis

The experimental design described above has three treatment factors: irradiance (L), species (S) and hydroponic nutrient concentration (N). We can therefore expand the allometric equation 2(b) as a general linear model that includes these three treatment factors, ln(MR), and all possible interactions, and test the statistical significance of each term in the model. In particular, our model had the form:

ln(ML) = a + bln(MR) + N + L + S + N · L +  N · S + L · S + S · ln(MR) +  N · ln(MR) + L · ln(MR) + N · L · S +  N · L · ln(MR) + N · S · ln(MR) +  L · S · ln(MR)N · L · S · ln(MR) + ɛ    eqn 4

The species term (S) is of no interest in differentiating between the balanced-growth and allometric allocation hypotheses. Interest lies in deviations of the allometric slope (b) from unity, as shown by a significant interaction between ln(MR) and either L or N, and in the significance of the N and L terms, which would show a difference in the allometric intercept across resource levels.

As it is possible that the allometric slope is not constant, we also used cubic spline smoothers (Shipley & Hunt 1996) to estimate the change in allometric slope and its 95% confidence intervals as a function of ln(ML); this was done by combining data for all species in a given irradiance and nutrient level. Statistical analyses were done using s-plus (SPLUS 1999).


Table 1 gives the median, first and third quartiles for the relevant biomass ratios in each treatment. Ranking the four treatments from highest to lowest for each variable shows how biomass allocation varied:

Table 1.  Median (first to third quartile) values of selected allocation parameters based on 1150 plants of 22 different species grown in factorial combinations of high (L, 1100 µmol m−2 s−1 PAR) and low (l, 200 µmol m−2 s−1 PAR) irradiance and high (N, full-strength Hoagland’s solution) and low (n, 1/6 dilution) nutrient supplies
Plant dry mass (g)0·26 (0·05–1·35)0·13 (0·03–0·43)0·16 (0·04–0·52)0·11 (0·03–0·29)
Leaf : plant dry mass (g g−1)0·50 (0·42–0·59)0·44 (0·25–0·52)0·53 (0·45–0·60)0·46 (0·36–0·54)
Root : plant dry mass (g g−1)0·33 (0·27–0·41)0·42 (0·35–0·50)0·27 (0·22–0·34)0·39 (0·33–0·47)
Stem : plant dry mass (g g−1)0·13 (0·06–0·22)0·09 (0·06–0·13)0·16 (0·09–0·18)0·12 (0·08–0·19)

for absolute growth the ranking was LN > lN > Ln > ln;

for allocation to leaves, lN > LN > ln > Ln;

for allocation to roots, Ln > ln > LN > lN;

for allocation to stems, lN > LN > ln > Ln.

Table 2 summarizes the anova of the generalized linear model. There were obvious differences between species, a clear allometric scaling between leaf and root biomass allocations, differences between species in the value of the allometric slope and, more importantly, differences in the allometric slope across environments. This latter effect was indicated by significant two-way interactions between ln(MR) and nutrients (N) and by a significant three-way interaction between ln(MR) and light (L) across species (S).

Table 2. anova of the generalized linear model relating ln(leaf mass, ML) to ln(root dry mass, MR), species (S), irradiance level (L) and nutrient level (N)
SourcedfMSF (P > F)
  1. df, Degrees of freedom; MS, mean square; F, Fisher’s F and its associated probability under the null hypothesis.

Irradiance (L) 1 28·50 183·94 (<10−15)
Nutrients (N) 1 151·40 977·29 (<10−15)
Species (S) 21 56·73 366·16 (<10−15)
ln(roots) (MR) 12160·8613948·21 (<10−15)
L : N 1  0·79  5·12 (0·024)
L : S 21  0·30  1·91 (0·008)
L : MR 1  0·37  2·40 (0·12)
N : S 21  0·62  3·99 (<10−9)
N : MR 1  2·23  14·40 (1·5 × 10−4)
S : MR 21  0·91  5·90 (<10−15)
L : N : S 19  0·29  1·87 (0·01)
L : N : MR 1  0·07  0·46 (0·50)
L : S : MR 21  0·44  2·82 (2·5 × 10−5)
N : S : MR 21  0·26  1·70 (0·03)
L : N : S : MR 19  0·19  1·21 (0·24)
Residuals978  0·15 

Combining all species together and fitting linearized allometric regressions of root dry mass (MR) on leaf dry mass (ML) for each treatment group, the following equations were obtained (standard errors in parentheses):

LN: ln(ML) = 0·17(±0·05) + 0·92(±0·01)ln(MR); SEest = 0·52; r2 = 0·94
Ln: ln(ML) = −0·48(±0·07) + 0·84(±0·02)ln(MR); SEest = 0·57; r2 = 0·87
lN: ln(ML) = 0·49(±0·06) + 0·96(±0·01)ln(MR); SEest = 0·45; r2 = 0·93
ln: ln(ML) = −0·28(±0·06) + 0·88(±0·02)ln(MR); SEest = 0·48; r2 = 0·90.

Figure 1 plots these regression equations to provide a visual evaluation of how the treatment environments changed the allometric allocation relationships. At the beginning of the experiment, plants had more leaf than root biomass. Relatively more biomass was subsequently allocated to roots than to leaves as the plants grew. This tendency was greater in the low-nutrient treatments than in the high-nutrient treatments. Within a nutrient level, this tendency was greater in the high-irradiance treatments than in the low-irradiance treatments. Table 3 lists the allometric slopes of the linearized allometric regressions obtained separately for each species; note that Eupatorium maculatum was not grown in the lN treatment and Hordeum jubatum was not grown in the Ln treatment. The number of species whose allometric slope was significantly less than 1·0 in each of the treatments was 10/22 (LN), 6/21 (Ln), 5/21(lN) and 10/22 (ln). In no case was the allometric slope significantly greater than 1·0.

Figure 1.

Allometric curves describing the growth of leaves and roots of plants growing in four different experimental environmental conditions: high (L, 1100 µmol m−2 s−1) and low (l, 200 µmol m−2 s−1 PAR) irradiance crossed with a full-strength (N) Hoagland’s hydroponic solution and a 1/6 dilution (n) of this solution. (a) Natural logarithmic axes: broken lines indicate the origin and thick solid line indicates a 1 : 1 relationship; (b) the same curves plotted on arithmetic axes.

Figure 2 plots the change in the allometric slope and its 95% confidence interval as a function of ln(ML) in each treatment combination based on cubic spline smoothers. An increase in root mass always resulted in a less than proportional increase in leaf mass – thus an allometric slope of less than unity – and this tendency increased as plants grew larger.

Figure 2.

Changing values of the slope (β) of the allometric relationship of ln(leaf mass) = α + β ln(root mass) for different values of ln(root mass), along with 95% confidence intervals of the slope, based on plants pooled within each experimental treatment: high (1100 µmol m−2 s−1) and low (200 µmol m−2 s−1 PAR) irradiance crossed with a full-strength (high) Hoagland’s hydroponic solution and a 1/6 dilution (low) of this solution. Broken horizontal lines show an allometric slope of 1·0 (balanced growth).


The allometric allocation hypothesis of Müller et al. (2000) is contradicted by our data in two ways. First, Müller et al. (2000) found preferential allocation to leaves over roots as plants grew larger, while we found the contrary. Second, our results show that the intercept and slope of the allometric relationship between leaves and roots varied as a function of the external supply of nutrients and light. On the other hand, the most general conclusion of Müller et al. (2000) does apply to our data: allocation often follows an allometric relationship whose slope is different from unity. This means that simple ratios will provide a biased measure of biomass allocation, as such ratios will necessarily vary with plant size.

The predictions of the balanced-growth hypothesis that we tested were supported by our data. Increasing the external nutrient supply when plants were in a high light environment increased the allocation of biomass to leaves, not only in absolute terms but also in terms of allometric allocation. This can be seen by comparing the Ln and LN allometric curves of Fig. 1. Similarly, increasing the irradiance when plants were in the high-nutrient environment increased the allocation of biomass to roots, not only in absolute terms but also in terms of allometric allocation. This can be seen by comparing the lN and LN allometric curves of Fig. 1. The allometric slope was generally less than unity, and this tendency to allocate proportionally more biomass to roots than to leaves increased as plants grew, rather than decreasing.

These results, while consistent with the balanced-growth hypothesis, should not be interpreted as a complete test of it. In order for the balanced-growth hypothesis to fit our data it must be true that the net uptake rate of nutrients per unit root mass (Um) decreased over time more than did the net photosynthetic rate per unit leaf mass (Am). There are at least two reasons to suspect this is true. First, the rate of nutrient uptake decreases as roots age, although this varies between ions (Clarkson 1996). As a greater proportion of the root system will consist of such older roots as plants age, the average net uptake rate of nutrients per unit root mass (Um) will decrease if the external supply rate is constant (Robinson 1986). Second, as the root system grew in the containers, the amount of nutrient solution would decrease more rapidly between flushes – whose frequency did not change – thus decreasing Um even further. As the plants were spaced so that there was minimal overlap of leaves between plants, the photon flux density experienced by leaves would not decrease as much over time as the decrease in nutrient supply rate. Although the intrinsic net photosynthetic rate of leaves does vary over time, this is likely to be less pronounced than variation in Um until leaves begin to senesce. It is therefore likely that the ratio Um/Am decreased in our experiment as plants grew larger, resulting in a proportionally larger allocation over time to roots rather than leaves. The fact that, at the end of the experiment, the allometric slope favoured roots more than leaves, and more so in the nutrient-limited environments (Ln and ln), is consistent with this explanation. In fact, the greater proportional allocation of biomass to roots than to leaves in plants growing alone in optimal (or near optimal) growth conditions appears to be a common pattern (Hunt & Lloyd 1987; Hunt & Nicholls 1986; Hunt, Nicholls & Fathy 1987; Shipley & Peters 1990).

We have no good explanation for the divergent results reported here with respect to Müller et al. (2000). Although those authors used more species (27 versus 22 in this study) they used fewer plants overall (562 versus 1150 in this study). It is possible that the difference is greater statistical power in the present study. However, there were many other differences as well. Müller et al. (2000) grew their plants for a longer time: 4 months versus 35 days in this study. The frequency with which the plants were given the hydroponic solution was much less – only 50 mL per plant per week – meaning that the temporal fluctuations in nutrient supply would have been much greater in their study. Finally, plants were grown together and it seems likely that, as plants grew larger, there would have been competition for light and nutrients that may have obscured the effects observed here. Clearly, more independent studies are needed in this area of research.

The assumption of balanced growth found in many ecological models involving plant growth, in which biomass is preferentially allocated to the plant part obtaining the resource that is limiting growth, appears to be a reasonable approximation based on our results. However, it is unlikely that plants growing in nature are ever truly at a dynamic equilibrium with respect to biomass allocation, as supply rates of light and soil resources fluctuate continuously. It is probably more appropriate to say that plants allocate biomass to different parts to reduce any imbalance between carbon fixation by leaves and soil resource acquisition by roots. Thus the degree of plasticity in changing these allometric allocations is probably important, and should be related to the temporal variability in resource supply rates in nature that different species typically experience.


This study was funded by the Natural Sciences and Engineering Research Council of Canada.