Scaling the leaf length-times-width equation to predict total leaf area of shoots.

BACKGROUND AND AIMS: An individual plant consists of different-sized shoots, each of which consists of different-sized leaves. To predict plant-level physiological responses from the responses of individual leaves, modelling this within-shoot leaf size variation is necessary. Within-plant leaf trait variation has been well investigated in canopy photosynthesis models but less so in plant allometry. Therefore, integration of these two different approaches is needed.
METHODS: We focused on an established leaf-level relationship that the area of an individual leaf lamina is proportional to the product of its length and width. The geometric interpretation of this equation is that different-sized leaf laminas from a single species share the same basic form. Based on this shared basic form, we synthesized a new length-times-width equation predicting total shoot leaf area from the collective dimensions of leaves that comprise a shoot. Furthermore, we showed that several previously established empirical relationships, including the allometric relationships between total shoot leaf area, maximum individual leaf length within the shoot and total leaf number of the shoot, can be unified under the same geometric argument. We tested the model predictions using five species, all of which have simple leaves, selected from diverse taxa (Magnoliids, monocots and eudicots) and from different growth forms (trees, erect herbs and rosette herbs). KEY
RESULTS: For all five species, the length-times-width equation explained within-species variation of total leaf area of a shoot with high accuracy (R2 > 0.994). These strong relationships existed despite leaf dimensions scaling very differently between species. We also found good support for all derived predictions from the model (R2 > 0.85).
CONCLUSIONS: Our model can be incorporated to improve previous models of allometry that do not consider within-shoot size variation of individual leaves, providing a cross-scale linkage between individual leaf-size variation and shoot-size variation.

Introduction

Plants are modular organisms, and they can be considered as a population of leaves and stems (Harper and Bell, 1979). Within each plant, organs (e.g. leaf or stem) usually differ in size, physiology and microenvironments (Field, 1983; DeJong ; Koyama and Kikuzawa, 2010; Niinemets, 2016; Kusi and Karsai, 2020; Maslova ). Therefore, photosynthesis of individual plants or ecosystems has been modelled as the sum of those of individual leaves (Bazzaz and Harper, 1977; Field, 1983; Ackerly and Bazzaz, 1995; Koyama and Kikuzawa, 2009). This cross-scale relationship between organ-level and plant- or ecosystem-level physiology has long been recognized as one of the central issues in canopy photosynthesis models (Field, 1991; Hikosaka ; Niinemets, 2016).

However, despite its importance, within-canopy or within-plant variation of organs has rarely been incorporated in the field of plant allometry. Allometry (i.e. power functions) has been a successful tool for analysing relationships between the properties of different-sized individual plants or organs (Niklas, 1994; Enquist ; Mori ; Savage ; Bentley ; Okie, 2013; Banavar ; Huang ; Lin ; Olson ; Kurosawa ; Wang ). However, most plant-level allometric models are based on the simplifying assumption that each individual plant has terminal organs (twigs or leaves) of the same size (Enquist ; West ; Savage ; Banavar ). These approaches contrast with organ-level studies on the within-plant size variation of twigs and leaves (Dombroskie and Aarssen, 2012; Koyama , 2017; Kusi and Karsai, 2020; Maslova ). The integration of these two approaches, plant allometry and canopy photosynthesis models, has not been achieved yet, although both approaches independently predict plant- or ecosystem-level metabolism (Koyama ).

Here, a shoot is defined as a terminal single current-year stem with all its appendages (leaves, buds, flowers, fruits, etc.). A shoot is equivalent to an individual ramet (i.e. whole above-ground part of a plant) in single-stem herbaceous species. For trees, a shoot is a fundamental unit of growth (Sterck ; Sterck and Schieving, 2007; Lecigne ) and reproduction (Chen ; Scott and Aarssen, 2013; Miranda ; Fajardo ). Given its importance, allometric relationships of shoot size and total shoot leaf area have been important topics in plant ecophysiology (Corner, 1949; White, 1983; Ackerly and Donoghue, 1998; Brouat ; Westoby and Wright, 2003; Kleiman and Aarssen, 2007; Olson , 2018; Sun , 2020; Yan ; Trueba ; Fan ; Smith ; Zhu ; Fajardo ). However, most previous studies on leaf vs. shoot size allometry have focused on the relationship among shoot size, total shoot leaf area, total leaf number and/or mean individual leaf size on the shoot. These studies are not mutually exclusive of, but do not yet have a theoretical connection with, the fact mentioned above that a shoot has a population of leaves with a size distribution (see Bazzaz and Harper, 1977). Because the total leaf area of a shoot (or a plant) is the sum of the areas of individual leaves, the leaf size distribution within a shoot is one of the main determinants of whole-plant or total shoot leaf area (Seleznyova and Greer, 2001; Bultynck ). Yet, with only a few exceptions (e.g. Koyama ; Smith ), this fact was not considered in most previous studies on leaf size – shoot size allometry (e.g. Sun , 2010, 2017, 2019, , 2020; Kleiman and Aarssen, 2007; Ogawa, 2008; Yang , 2009, 2010; Milla, 2009; Xiang , 2010; Whitman and Aarssen, 2010; Dombroskie and Aarssen, 2012; Scott and Aarssen, 2012, 2013; Yan ; Dombroskie ; Trueba ; Olson ; Miranda ; Zhu ; Fajardo ).

Therefore, the objective of this study was to clarify the relationship between size variations at two different levels: the within-species size variation of shoots and the within-shoot size variation of leaves. We propose a simple geometric model that incorporates these two size variations. The model is a mathematical quantification and generalization of the results of Koyama , which showed that differently sized plants of the herbaceous species Cardiocrinum cordatum share the same basic structure. However, their study did not provide a mathematical model that could derive these relationships. Furthermore, the present model is more general than the findings of Koyama , in that it can be applied to various plant forms (trees, rosettes and erect herbs). In the present model, maximum leaf size within a shoot plays a pivotal role. In relation to this, Sun , 2020) recently proposed a model that unified previous studies on the leaf size–number trade-off (Kleiman and Aarssen, 2007), shoot photosynthesis and growth (Niklas and Enquist, 2001, 2002), and stem cross-sectional area [i.e. pipe model (Shinozaki ; Brouat )]. Sun , 2020) also found that maximum leaf size within a shoot is a major determinant of the leaf number per stem mass across different species. Moreover, Lopes and Pinto (2005), and Heerema, Spann, and their colleagues (Heerema ; Spann and Heerema, 2010) proposed empirical relationships that use maximum leaf size to predict total shoot leaf area. Nonetheless, all of these previous findings, specifically on the usefulness of maximum leaf size, are phenomenological because they do not provide any quantitative model to explain why maximum leaf size is a predictor of the total leaf area of a shoot. Here, we used an entirely novel approach, which uses maximum leaf size to model within-shoot and between-shoot leaf size variations.

MODEL

Individual leaf area (Aleaf) is defined as the area of one side of each lamina (i.e. leaf blade) (John ). A shoot may have one or multiple leaves, each of which may differ in size. Therefore, the total leaf area of a shoot (Ashoot) is defined as the sum of Aleaf of all the leaves on that shoot:

The symbol ‘≡’ indicates ‘defined as’. As our aim was to find simple formulas that predict Ashoot, taking into consideration the within-shoot size variation of Aleaf, the present model is based on several simplifications. (1) We focused only on the leaf laminas that determine Ashoot. We thus ignored any other organs (e.g. stem, petioles, buds and reproductive organs). (2) Our model only deals with simple leaves with flat-shaped laminas: the current model cannot be applied to leaves of different forms (e.g. compound leaves that consist of multiple leaflets, succulent leaves or conifer needles). The limitations associated with these simplifications will be addressed in the Discussion.

We use the two words ‘similar’ and ‘affine’ (Fig. 1), which have been used as compound words ‘self-similar’ and ‘self-affine’ in fractal geometry (Falconer, 2003; Okie, 2013; Shi ). In Fig. 1, in each panel (A and B), the two green triangles represent two different-sized individual leaf laminas. Two shapes are similar if they can be made identical by multiplying each dimension by a single constant (i.e. similar transformation). Two shapes are affine if they can be made identical by multiplying each dimension by a different constant (i.e. affine transformation).

Fig. 1.

Definition of the words ‘similar’ and ‘affine’ used in this article. (A) Two similar triangles share the same length-to-width ratio. (B) Two affine triangles may have different length-to-width ratios. For an affine transformation to change the small triangle into the large triangle, the scaling factor in one direction (a) is not necessarily equal to that in the other direction (b), and similar transformation is a special case of affine transformation when a = b. In both cases, the area is proportional to the product of the length (L) and width (W).

First, we focused on individual leaves. Within a species, the area of an individual leaf (Aleaf) is proportional to the lamina length (Lleaf) times lamina width (Wleaf) (Cain and Castro, 1959; Teobaldelli , ; Yu ; Huang ; Li ; Schrader ; Shi ) (Fig. 2A):

Fig. 2.

The length-times-width model for (A) an individual leaf and (B) a shoot.

The symbol ‘’ indicates ‘proportional to’. Equation (2) is known as the Montgomery equation (Yu ; Shi ). It indicates that leaves from the same species are affine to each other.

Next, we extend eqn (2) to the level of shoots to predict Ashoot. We hypothetically detach all the leaf laminas from the stem, and place them side-by-side on a flat plane to determine its dimensions as illustrated in Fig. 2B:

We refer to this set of leaf laminas as the ‘foliage’ of each shoot. The subscripts ‘f’ in eqn (3) stand for ‘foliage’. We exclude petioles because they contribute to the 3D arrangement with relatively little contribution to Ashoot. The utility of rearranging the leaves is that both foliage length (Lf) and foliage width (Wf) can be defined independently from the 3D arrangement of the leaves. Our main hypothesis is that, within a single species, different-sized sets of foliage are affine, as is the case of individual leaves. This indicates that the total area of a foliage (which is Ashoot, by definition) is proportional to the product of Lf and Wf (Fig. 2B):

We will call eqn (4) the ‘foliage length-times-width equation’. Note that foliage length (Lf) is defined using leaf width (not leaf length), and foliage width (Wf) is defined using leaf length. The reason for these definitions is that both ‘leaf length’ and ‘foliage length’ are defined in the proximal–distal direction. In other words, foliage is analogous to a pinnately compound leaf that extends from the parent stem in the distal direction.

Next, we compare different-sized shoots from a single species. As shoot size increases, both Lf and Wf increase. We assume that the ratio of the relative growth rates of the foliage in these two directions is constant (Huxley, 1932; Niklas, 1994; Okie, 2013), and therefore follows the allometric relationship:

The exponent β is expected to be >1, for the following reason. If foliage always consists of a single leaf, irrespective of its size, by definition Lf and Wf are equivalent to Wleaf and Lleaf, respectively. In this case, Lf and Wf should be approximately proportional to each other (i.e. β ≈ 1). However, in reality, a shoot usually has multiple leaves. Because Lf is defined as the sum of the widths of all leaves, larger foliage with more leaves should have a disproportionately larger length relative to its width than small foliage (β > 1). In general, the value of β may vary among species, depending on the species’ intrinsic maximum leaf size and leafing intensity. In the Results, we show that eqn (5) is valid. Before demonstrating this, we proceed by assuming that eqn (5) is valid to derive other predictions. By combining eqns (4) and (5), we obtained:

As mentioned above, the lamina area of an individual leaf is predicted by the product of lamina length and width with high accuracy (i.e. high R2 values). Additionally, it is known that individual leaf area can also be predicted by a quadratic function of lamina length or width alone [e.g. Aleaf ∝ (Lleaf)2], albeit with less accuracy (Teobaldelli , ). Similarly, eqn (6) predicts that Ashoot can also be predicted by Wf alone, with less but acceptable accuracy. Note that because β > 1, the exponent is expected to be >2. Suppose we further use an empirical relationship that individual lamina length is approximately proportional to lamina width [Lleaf ∝ Wleaf (Ogawa )], by using eqn (6), we predicted that the maximum leaf lamina width within a shoot can also be used as a predictor of Ashoot:

These predictions [eqns (6) and (7)] were also tested in this study. Previous studies have already recognized the usefulness of maximum leaf size as a predictor of Ashoot (Lopes and Pinto, 2005; Heerema ; Sun ; Teobaldelli ). However, these studies used maximum leaf size only as empirical models. Therefore, none of them has provided a quantitative theory that explains why this relationship holds. In the following subsections, we show that these empirical relationships can also be derived as corollaries of the present model.

Sun et al.’s equation

Sun , 2020) found that Ashoot is proportional to the product of the maximum leaf area and total number of leaves on each shoot (N), because the maximum individual leaf area of a shoot corresponds to its potential leaf-producing capacity. This relationship can also be derived from our model (see Appendix for derivation):

We retest this prediction in this study.

Size–number allometry

We also derived the allometric relationship between Ashoot and the total number of leaves on each shoot (N) reported by Koyama (see Appendix for derivation):

Generally, the exponent α may vary depending on species as a function of β. The predicted allometric relationship between Ashoot and N with the exponent α > 1 (given β > 1) agrees with the empirical result reported by Koyama . We retested this prediction in this study. In addition, eqn (9) can be rearranged to predict the scaling relationship between mean individual leaf area (=Ashoot/N) and Ashoot with the exponent 0 < λ < 1 (see Appendix for derivation):

The prediction that 0 < λ < 1 was empirically supported by Smith .

Heerema–Spann–Teobaldelli et al.’s equation

Heerema, Spann and their colleagues (Heerema ; Spann and Heerema, 2010) reported an empirical relationship that Ashoot can be predicted by the maximum leaf length of a shoot (i.e. foliage width, Wf) and the total number of leaves on that shoot (N) using woody fruit crop species. Teobaldelli modified this relationship into a general allometric form. These relationships can also be derived from our model (see Appendix for derivation):

Equation (11) was proposed as an empirical model by Teobaldelli , which includes the formula proposed by Heerema, Spann and their colleagues as a specific case when γ = 1, which does not take into consideration the β-dependency of γ. Generally, γ may vary among species as a function of β. Here, eqn (11) was tested by the following allometric relationship:

We also directly tested eqn (12). Unlike eqns (4) and (5), eqn (12) does not use foliage length (Lf) as a variable, and therefore eqn (12) can be tested independently.

Lopes–Pinto’s equation

Lopes and Pinto (2005) found an empirical formula that predicts Ashoot for a wine grape variety using the maximum and minimum leaf area within each shoot. They found that each shoot’s mean individual leaf area can be estimated as the mean of maximum and minimum leaf area within that shoot. This relationship can also be derived from our model (see Appendix for derivation):

The symbol k is a proportionality constant. Lopes, Pinto and colleagues (Lopes and Pinto, 2005; Phinopoulos ) found the same relationship as eqn (13) for two wine grape varieties as an empirical formula. They used an empirical value of k = 1 (i.e. in their cases, mean individual leaf area was simply the average value of the largest and the smallest leaves) as a specific value for the grape varieties. Generally, k may vary depending on the species (depending on the arrangement of different-sized leaves along a shoot). This prediction was also tested in this study.

MATERIALS AND METHODS

Study species

The study species and the sample sizes are listed in Table 1. Each species is referred to by its genus name after its first mention. All species have simple leaves with reticulate or reticulate-like venation patterns. (1) Kobushi magnolia (Magnolia kobus, Magnoliaceae). Magnolia was selected because it is taxonomically separate from the other species (APG IV, 2016). (2) Cardiocrinum cordatum (including var. glehnii) (Liliaceae) is a monocarpic perennial herb. This species belongs to the monocots (APG IV, 2016), but its leaves have reticulate venation patterns that are similar to those of eudicots (see photographs in Koyama ). Small individual plants form rosettes on the ground without elongating their stems, whereas large plants become bolting rosettes, which elongate their vertical stems with flower buds on top (Ohara ; Komamura ). (3) Sargent’s cherry (Prunus sargentii, Rosaceae) and (4) Japanese elm (Ulmus davidiana var. japonica, Ulmaceae). Prunus and Ulmus were chosen as typical broadleaved deciduous trees in temperate forests. (5) Giant knotweed (Fallopia sachalinensis, Polygonaceae) is a high-stature erect herb (plant height often reaches 2–3 m) with large leaves along its vertical stem. Cardiocrinum and Fallopia were chosen because they have contrasting growth forms (rosette vs. erect) and are from different taxonomic groups (monocot vs. eudicot).

Table 1.

Study species and sample sizes

			Magnolia kobus	Cardiocrinum cordatum	Prunus sargentii	Ulmus davidiana var. japonica	Fallopia sachalinensis
Taxonomy			Magnoliid (Magnoliales, Magnoliaceae)	Monocot (Liliales, Liliaceae)	Eudicot (Rosales, Rosaceae)	Eudicot (Rosales, Ulmaceae)	Eudicot (Caryophyllales, Polygonaceae)
Growth form			Tree (deciduous)	Herb (rosette or bolting)	Tree (deciduous)	Tree (deciduous)	Erect herb
Location			R, T	F, H	U	U, F	U
Number of shoots investigated			37	36	39	43	29
Size ranges	A shoot (cm2)	min	11.3	11.6	9.5	1.2	62.3
		max	1440.8	5718.2	1884.6	652.3	11 716.5
	N	min	2	1	1	1	4
		max	11	22	20	15	19

Location of sampling: F: The Forest of Obihiro; H: natural forest preservation of Hokkaido Obihiro Agricultural High School; R: Urikari River; T: Tokachi Ecology Park; U: Obihiro University of Agriculture and Veterinary Medicine. Ashoot: total leaf area of each shoot (cm2); N: total leaf number of each shoot.

Field sampling

A shoot is defined herein as a single current-year stem with its appendages (leaves, buds, flowers, fruits, etc.). For the two single-stemmed herbaceous species (Cardiocrinum and Fallopia), a shoot is equivalent to an entire above-ground part of an individual ramet, and therefore Ashoot is equivalent to whole-plant leaf area. Sample sizes and the sampling locations are given in Table 1. Sampling was conducted in summer (June–August) in 2016 and 2020. All sampling sites were located in Obihiro City or the adjacent Otofuke Town in Hokkaido Island in a cool-temperate region of Japan, and were within 10 km from the Obihiro Weather Station (42°52′N 143°10′E, altitude: 76 m a.s.l.). Mean annual temperature and precipitation at the weather station during 1998–2017 were 7.2 °C and 937 mm, respectively (Japan Meteorological Agency, 2020). Shoots with obvious damage (e.g. leaf loss due to herbivory etc.) were excluded. For the woody species, shoots that had sylleptic shoots (i.e. branching within the current year) were not sampled. Our sampling strategy was not random, but instead the shoots were sampled to cover a wide range sizes within each species (i.e. small, medium and large shoots were intentionally selected). Because healthy shoots were selected based solely on their sizes, both shaded and well-lit shoots were sampled for trees. For herbaceous species (Cardiocrinum and Fallopia), all shoots (ramets) within the same species grew in similar environments in their natural habitats. Cardiocrinum were sampled in partially shaded forest understories or small gaps and Fallopia were sampled in open clearings. For Cardiocrinum, leaf sizes were measured non-destructively in situ (see below). For the other species, shoots were harvested using pruning scissors or a long-reach pruner, sometimes with the aid of a stepladder. Immediately after sampling, shoots were stored in closed plastic bags with wet paper towels to avoid desiccation. Scanning (described below) was conducted within the same sampling day.

Leaf size measurements

Leaf length (Lleaf) is defined as the length of the leaf lamina, measured from the lamina tip to the point at which the lamina attaches to the petiole. Leaf width (Wleaf) is defined as the maximum lamina width perpendicular to the midvein. Individual leaf area (Aleaf) is defined as the area of one side of each lamina (John ). For Cardiocrinum, we measured Lleaf and Wleaf of all leaves on each stem using a measuring tape in situ. Then, Aleaf for this species was estimated using the following equation: individual leaf area = 0.7169 (leaf length × width) (Koyama ). For the other species, the harvested leaves were scanned using flatbed digital scanners (LiDE 210, Canon, Tokyo, Japan, 400 dpi; or 400-SCN025, Sanwa Supply, Okayama, Japan, 600 dpi). The sizes (Lleaf, Wleaf, Aleaf) of each leaf were measured using ImageJ v.1.50i or 1.53a (Schneider ). For Cardiocrinum, both reproductive (large bolting plants) and vegetative shoots (rosettes) were sampled to cover the natural size range of this species, and the flower buds on top of Cardiocrinum stems were excluded as leaves. No reproductive organs were found among the sampled shoots of the other species. Some large shoots of Prunus and Ulmus trees, and most of the shoots of the erect herb Fallopia, were still elongating at the time of harvesting (June–August). For these shoots, only leaves of which laminae were unfolded (even when they were young and still expanding) were counted and measured; small folded immature leaves or leaf primordia near or at the shoot apical meristem were excluded as leaves. Among large shoots of Fallopia, small leaves were occasionally found on small lateral shoots that were branched from the main stem. These small lateral shoots were not measured because we focused on a single stem in this study. The total amount of those immature and lateral leaves was small compared to the total amount of leaves on the main stem.

All statistical analyses were performed with the statistical software R v.4.1.0 (R Core Team, 2021) and the packages cowplot (Wilke, 2016), ggplot2 (Wickham, 2016), gridExtra (Auguie, 2017) and smatr (Warton ). Following Warton , ordinary least squares (OLS) and/or standardized major axis (SMA) regression analyses were performed for each relationship. OLS lines were fitted to predict variable Y (e.g. Ashoot) from X (e.g. Lf × Wf) with the R function lm. SMA lines were fitted to determine the mutual allometric relationship between two variables (e.g. foliage length vs. width) with the sma function of the package smatr. The R2 values of the OLS lines reported in this article were adjusted.

RESULTS

For all species investigated, the foliage length-times-width equation (eqn 4) explains Ashoot with high accuracy (R2 > 0.994 for all species; Fig. 3; Table 2). As predicted by Eqns (6) and (7), Ashoot can also be predicted as an allometric equation of maximum leaf length (i.e. foliage width Wf; Fig. 4; Table 2) or maximum leaf width alone (Fig. 5; Table 2), though with less accuracy (OLS: R2 = 0.930–0.976; SMA: 0.932–0.977). The allometric relationship between foliage width and length (eqn 5) was also supported (R2 = 0.905–0.960; Fig. 6; Table 2). As expected, the scaling exponent β was >1, and the value of β varied greatly among the species (Table 2).

Fig. 3.

The total leaf area of a shoot (Ashoot) is proportional to the product of foliage length (Lf) and width (Wf), as predicted by eqn (4). Each closed circle indicates an individual shoot. See Fig. 2B for the definition of foliage length and width. The blue lines show OLS (ordinary least squares) regression lines (R2 > 0.994). See Table 2 for the regression results.

Table 2.

Results of the regression analyses (OLS: ordinary least squares; SMA: standardized major axis). All regressions are significant (P < 1.0 × 10 − 5 for all cases).

Y = a  + bX		Type		Mgk	Cac	Prs	Udj	Fas
Y	X
A_shoot	L_f × W_f{L_f≡∑shootW_leafW_f≡maxshoot(L_leaf)	OLS	a	−1.492	117.352	7.904	−2.109	−90.461
			b	0.546	0.543	0.519	0.562	0.652
			R 2	0.997	0.996	0.996	0.997	0.994
log_10(A_shoot)	log_10⁡W_f=log_10[maxshoot(L_leaf)]	OLS	a	−1.135	−1.758	−1.620	−0.565	−0.065
			b	3.088	3.554	3.682	2.647	2.511
			R 2	0.976	0.957	0.966	0.951	0.944
		SMA	a	−1.178	−1.866	−1.690	−0.606	−0.151
			b	3.125	3.630	3.745	2.712	2.582
			R 2	0.977	0.958	0.967	0.953	0.946
log_10(A_shoot)	log_10[maxshoot(W_leaf)]	OLS	a	−0.350	−0.502	−0.603	−0.199	−0.598
			b	3.084	2.841	3.654	3.064	3.111
			R 2	0.963	0.930	0.946	0.956	0.967
		SMA	a	−0.402	−0.635	−0.687	−0.228	−0.657
			b	3.141	2.943	3.754	3.132	3.162
			R 2	0.964	0.932	0.947	0.957	0.968
log_10⁡L_f	log_10⁡W_f	SMA	a	−0.939	−1.783	−1.387	−0.415	0.090
			b (=β)	2.187	2.856	2.806	1.892	1.629
			R 2	0.961	0.916	0.939	0.907	0.909
log_10(A_shoot)	log_10⁡N	OLS	a	0.742	1.853	0.830	0.333	0.103
			b	2.325	1.443	1.865	1.980	3.137
			R 2	0.963	0.857	0.946	0.855	0.922
		SMA	a	0.711	1.747	0.785	0.271	−0.013
			b (=α)	2.368	1.555	1.916	2.136	3.263
			R 2	0.964	0.861	0.948	0.859	0.925
A_shoot	N⋅maxshoot(A_leaf)	OLS	a	24.144	420.913	33.972	−4.077	−97.621
			b	0.630	0.369	0.637	0.690	0.712
			R 2	0.980	0.976	0.987	0.996	0.989
log_10(A_shootW_f)	log_10⁡N	OLS	a	0.111	0.796	0.137	−0.029	−0.065
			b	1.605	1.080	1.390	1.288	1.997
			R 2	0.977	0.896	0.962	0.871	0.943
		SMA	a	0.097	0.740	0.114	−0.064	−0.118
			b (=γ)	1.623	1.139	1.416	1.378	2.054
			R 2	0.977	0.899	0.963	0.874	0.945
A_shoot	N[minshoot(A_leaf)+maxshoot(A_leaf)2]	OLS	a	7.376	310.017	−16.200	−5.339	−122.219
			b (=k)	1.092	0.758	1.109	1.193	1.221
			R 2	0.994	0.981	0.991	0.992	0.993
A_leaf	L_leaf⋅W_leaf	OLS	a	1.163	–	−0.799	0.121	1.514
			b	0.648	–	0.643	0.663	0.798
			R 2	0.989	–	0.992	0.992	0.992

Mgk: Magnolia kobus; Cac: Cardiocrinum cordatum; Prs: Prunus sargentii; Udj: Ulmus davidiana var. japonica; Fas: Fallopia sachalinensis.

Fig. 4.

The total leaf area of a shoot (Ashoot) is a power function of foliage width (Wf), defined as the maximum individual leaf length of the shoot, as predicted by eqn (6). Each closed circle indicates one individual shoot. The blue lines show the OLS (ordinary least squares) regression lines (R2 = 0.944–0.976). See Table 2 for the regression results.

Fig. 5.

The total leaf area of a shoot (Ashoot) is a power function of the maximum individual leaf width of the shoot, as predicted by eqn (7). Each closed circle indicates an individual shoot. The blue lines show the OLS (ordinary least squares) regression lines (R2 = 0.930–0.967). See Table 2 for the regression results.

Fig. 6.

Log–log linear (allometric) relationship between foliage length (Lf) and width (Wf). The regression slopes correspond to β in eqn (5). Each closed circle indicates an individual shoot. The red lines show the SMA (standardized major axis) regression lines (R2 = 0.905–0.960). See Table 2 for the regression results.

As predicted by eqn (9), Ashoot is expressed as a power function of the total number of leaves on that shoot (N) with the SMA regression exponents >1 (Fig. 7; Table 2), though for this relationship substantial deviations from the regression lines (R2 = 0.859–0.964; Table 2) were observed in the region for N ≤ 3 (log10N ≤ 0.48). This is especially evident when N = 1, in which case Ashoot is represented by only a single leaf, and as N increases, the values of Ashoot become stable as they are calculated as the sum of many leaves. The present data also reconfirm all previously known empirical relationships found by Sun et al. [eqn (8); R2 > 0.976; Fig. 8; Table 2], by Heerema–Spann–Teobaldelli et al. [eqn (12); OLS: R2 = 0.871–0.977; SMA: R2 = 0.874–0.977; Fig. 9; Table 2] and by Lopes–Pinto [eqn (13); R2 > 0.981; Fig. 10; Table 2]. The present data also reconfirm the leaf-level relationships that individual leaf area is proportional to the product of its lamina length and width [eqn (2); R2 > 0.989; Table 2].

Fig. 7.

Log–log linear (allometric) relationship between the total leaf area of a shoot (Ashoot) and the total number of leaves on the shoot (N). The regression slopes correspond to α in eqn (9). Each closed circle indicates one individual shoot. The red lines show the SMA (standardized major axis) regression lines (R2 = 0.859–0.964). See Table 2 for the regression results.

Fig. 8.

The total leaf area of a shoot (Ashoot) is proportional to the product of the maximum individual leaf area and the number of leaves on the shoot (N), as predicted by eqn (8). Each closed circle indicates one individual shoot. The blue lines show the OLS (ordinary least squares) regression lines (R2 > 0.976).

Fig. 9.

The total leaf area of a shoot divided by foliage width (Ashoot/Wf) is a power function of the number of leaves on the shoot (N). The regression slopes correspond to γ in eqn (12). Each closed circle indicates an individual shoot. The blue lines show the OLS (ordinary least squares) regression lines (R2 = 0.871–0.977). See Table 2 for the regression results.

Fig. 10.

The total leaf area of a shoot (Ashoot) is proportional to the product of the number of leaves (N) times (maximum + minimum individual leaf area) divided by 2. The regression slopes correspond to k in eqn (13). Each closed circle indicates an individual shoot. The blue lines show the OLS (ordinary least squares) regression lines (R2 > 0.981). See Table 2 for the regression results.

Discussion

Leaf vs. shoot elongation

The structure of a shoot, including size variation and arrangement of leaves, determines the light-harvesting efficiency of plants (Givnish, 1984; Valladares and Brites, 2004; Pearcy ; Smith ; Olson ; Koyama ; Iwabe ). If a shoot is to minimize the cost of current light harvesting, the optimal solution derived by Givnish (1982) is to have a single large leaf with no investment in the stem (i.e. no stem elongation). Why does a shoot have multiple leaves instead of a single large leaf? There are mutually non-exclusive explanations for the benefit of producing multi-leaved stems rather than single-leaved stems. First, plants are subject to competition with neighbours (Givnish, 1982; Anten, 2016), and existing leaves will be gradually shaded by neighbouring plants in the future. Under competition, plants should continuously elongate their stems and produce new leaves in better-lit positions (Koyama and Kikuzawa, 2009; Anten, 2016; Koyama ). Therefore, a shoot has at least two functions in terms of light capture: current light harvesting and space acquisition, the latter of which contributes to future light harvesting (Yagi and Kikuzawa, 1999; Sterck ; Laurans and Vincent, 2016; Koyama ). Differentiation of short vs. long shoots can be considered as a continuum of a strategy along the trade-off between these two functions (Yagi and Kikuzawa, 1999). In the present dataset, the exponent β was >1 [eqn (5); Table 2], indicating that larger foliage had a larger foliage length relative to its foliage width, as expected because larger foliage consists of more leaves than smaller foliage (Fig. 7). This phenomenon is called geometric dissimilitude and it can be considered as a shift in strategy along size variation (Niklas, 1994; Okie, 2013). These results are consistent with the observation that long shoots are specialized for space acquisition whereas short shoots are specialized for light capture, and there is a continuous shift between these two extremes (Yagi and Kikuzawa, 1999). Second, larger leaves produce a thicker boundary layer that reduces heat and gas exchange (Schuepp, 1993; Xu ); therefore, larger leaves are subject to greater heat stress (Vogel, 2009). Having compound leaves that consist of multiple leaflets instead of simple large leaves can effectively reduce the boundary layer resistance (Gurevitch and Schuepp, 1990; Xu ). At the level of individual leaves, Schrader demonstrated that the length-times-width equation (eqn 2) is valid for compound leaves. However, at the shoot level, leaf shape (e.g. simple vs. compound) may also affect the leaf–shoot allometric relationship (Yang ). Therefore, the scaling relationships may also be affected by the leaf shape, which is in turn is affected by the environment (Royer ; Xu ). Third, Kleiman and Aarssen (2007) suggested that producing more leaves, instead of fewer but larger ones, is more beneficial because it allows stems to have more buds and eventually leads to greater lateral growth and higher plasticity of allocation between growth and reproduction. Fourth, for a given limit on the total leaf area of a shoot, larger leaves incur a disproportionately greater cost of supporting tissues (Niinemets ; Shi ). Fifth, if a plant has many leaves, then the feeding or attacking efficiency of herbivores or pathogens may be reduced (Brown ). Altogether, the observed variation of β across the five species may reflect these multiple compounding factors. Therefore, further investigations on species with different leaf shapes (such as compound leaves), leaf sizes, leafing intensities and environments (including herbivores and pathogens) are needed.

In this study, we intentionally ignored the 3D arrangement of foliage and instead considered the 2D structure of foliage as being analogous to a single large leaf (Fig. 2B). By doing so, our length-times-width equation successfully predicted Ashoot with high accuracy without considering any details of the actual foliage structure other than size. The simplification applied in this study is in contrast to existing models, which consider the 3D arrangement of leaves, such as phyllotaxis (Valladares and Brites, 2004; Smith ), internode length (Meng ), stem inclination angle (Meng ), and the resultant light interception and within-shoot self-shading (Valladares and Brites, 2004; Koyama and Kikuzawa, 2010; Smith ; Olson ). Our model does not consider stem traits, such as cross-sectional area (Brouat ; Yan ; Smith ; Lehnebach ; Sun , b, 2020; Fajardo ), length-to-diameter ratio (Xiang , Levionnois ), conduit size, which determines hydraulic efficiency (Savage ; Chen ; Fan ; Trueba ; Olson ; Bortolami ; Levionnois ), stem mechanical properties (Brouat and McKey, 2001; Chen ; Trueba ; Fan ; Olson ; Baer ; Levionnois ) or the associated stem construction costs (Yang ; Givnish, 2020). Nonetheless, because our model focuses only on a population of leaf laminas, it is not mutually exclusive to the previous models. Instead, the geometric property of foliage can be incorporated to improve the previous models, which do not consider the within-shoot size variation of individual leaves.

Limitations of the model

The product of leaf lamina length and width can predict individual leaf area, and this relationship holds for diverse taxa and for different growth conditions, without considering the underlying leaf structures such as venation (Blonder ; Kawai and Okada, 2020), lobation (Schuepp, 1993; Kusi and Karsai, 2020), lamina folding (Fleck ; Deguchi and Koyama, 2020), epidermal features (Maslova ) and internal mesophyll structures (Oguchi ), all of which are known to differ among angiosperm species and under different environmental conditions. The consistency of the shoot-level results among eudicots, Magnoliids and monocots obtained herein may imply that the present results may also be generalized across angiosperms, as is the case for individual leaves. However, because our aim was to propose and test a new model as a starting point, we chose only five typical temperate woody and herbaceous species. In general, leaf–shoot allometric relationships are affected by climate or altitude (Westoby and Wright, 2003; Sun , 2019; Xiang , , 2010; Zhu ), as well as by leaf habit (i.e. deciduous vs. evergreen) (Brouat ; Yang , 2009; Milla, 2009; Zhu ; Fajardo ). Therefore, it remains unclear whether the present results can be applied to different situations, including other species from extreme climates or different life forms, such as evergreen conifers. Additionally, our model does not consider compound leaves. At the level of individual leaves, a recent study demonstrated that the length-times-width equation (eqn 2) is valid for both simple and compound leaves (Schrader ). However, at the level of shoots, leaf shape may also affect the leaf–shoot allometric relationship (Yang ). Therefore, more comprehensive datasets that include a diversity of leaf forms are needed to validate our model. Furthermore, our model does not consider the reproductive organs. The scaling relationships between reproductive organs and shoot size has long been recognized (Chen ; Scott and Aarssen, 2013; Miranda ), and the existence of reproductive organs also alters scaling relationship among vegetative organs (Fajardo ). Therefore, future studies are needed to elucidate whether the simple relationship found in the present study is affected by the existence of reproductive organs.

Conclusions

Based on the geometric properties of foliage, we proposed the ‘foliage length-times-width equation’ that accurately predicts the total leaf area of a shoot. The model unifies several previously established empirical relationships into a single theory. We also demonstrated that the total leaf area of a shoot can also be predicted by maximum individual leaf lamina length or width alone. The dataset of five species from diverse taxa generally supported the model predictions, though deviations from the model were also observed. More comprehensive datasets that include a diversity of species are needed to test the generality of our model in future studies.