## Introduction

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 *M*_{L} and *M*_{R} are the dry mass of photosynthetic organs (leaves) and roots; *A*_{m} and *U*_{m} 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:

*M*

_{L}·

*A*

_{m}∝

*M*

_{R}·

*U*

_{m}eqn 1a

Rearranging equation 1a and taking logarithms, we obtain:

*M*

_{L}) ∝ ln(

*M*

_{R}) + ln(

*U*

_{m}) − ln(

*A*

_{m}) eqn 2a

*M*

_{L}) = α + βln(

*M*

_{R}) + δ

_{1}ln(

*U*

_{m}) − δ

_{2}ln(

*A*

_{m}) eqn 2b

The partial intercept (α) and partial slope β quantify the allometric relationship between leaf and root mass, and the partial slopes δ_{1} and δ_{2} 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 (*M*_{R}). 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 (*A*_{M}).

### 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:

*M*

_{R}) =

*a*+

*b*ln(

*M*

_{L}) +

*N*+

*N ·*ln(

*M*

_{L}) + 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 (*U*_{m}). 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(*M*_{L}) 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.