Leaf dry mass per unit area (LMA) is considered to represent the photosynthetic capacity, which actually implies a hypothesis that foliar water mass (leaf fresh weight minus leaf dry weight) is proportional to leaf dry weight during leaf growth. However, relevant studies demonstrated that foliar water mass disproportionately increases with increasing leaf dry weight. Although scaling relationships of leaf dry weight vs. leaf area for many plants were investigated, few studies compared the scaling relationship based on leaf dry weight with that based on leaf fresh weight. In this study, we used the data of three families (Lauraceae, Oleaceae, and Poaceae, subfamily Bambusoideae) with five broad-leaved species for each family to examine whether using different measures for leaf biomass (i.e., dry weight and fresh weight) can result in different fitted results for the scaling relationship between leaf biomass and area. Reduced major axis regression was used to fit the log-transformed data of leaf biomass and area, and the bootstrap percentile method was used to test the significance of the difference between the estimate of the scaling exponent of leaf dry weight vs. area and that of leaf fresh weight vs. area. We found that there were five species across three families (Phoebe sheareri (Hemsl.) Gamble, Forsythia viridissima Lindl., Osmanthus fragrans Lour., Chimonobambusa sichuanensis (T.P. Yi) T.H. Wen, and Hibanobambusa tranquillans f. shiroshima H. Okamura) whose estimates of the scaling exponent of leaf dry weight to area and that of leaf fresh weight to area were not significantly different, whereas, for the remaining ten species, both estimates were significantly different. For the species in the same family whose leaf shape is narrow (i.e., a low ratio of leaf width to length) the estimates of two scaling exponents are prone to having a significant difference. There is also an allometric relationship between leaf dry weight and fresh weight, which means that foliar water mass disproportionately increases with increased leaf dry weight. In addition, the goodness of fit for the scaling relationship of leaf fresh weight vs. area is better than that for leaf dry weight vs. area, which suggests that leaf fresh mass might be more able to reflect the physiological functions of leaves associated with photosynthesis and respiration than leaf dry mass. The above conclusions are based on 15 broad-leaved species, although we believe that those conclusions may be potentially extended to other plants with broad and flat leaves.

Plant leaves exhibit diverse shapes that enable them to utilize a light resource maximally. If there were a general parametric model that could be used to calculate leaf area for different leaf shapes, it would help to elucidate the adaptive evolutional link among plants with the same or similar leaf shapes. We propose a simplified version of the original Gielis equation (SGE), which was developed to describe a variety of object shapes ranging from a droplet to an arbitrary polygon. We used this equation to fit the leaf profiles of 53 species (among which, 48 bamboo plants, 5 woody plants, and 10 geographical populations of a woody plant), totaling 3310 leaves. A third parameter (namely, the floating ratio c in leaf length) was introduced to account for the case when the theoretical leaf length deviates from the observed leaf length. For most datasets, the estimates of c were greater than zero but less than 10%, indicating that the leaf length predicted by the SGE was usually smaller than the actual length. However, the predicted leaf areas approximated their actual values after considering the floating ratios in leaf length. For most datasets, the mean percent errors of leaf areas were lower than 6%, except for a pooled dataset with 42 bamboo species. For the elliptical, lanceolate, linear, obovate, and ovate shapes, although the SGE did not fit the leaf edge perfectly, after adjusting the parameter c, there were small deviations of the predicted leaf areas from the actual values. This illustrates that leaves with different shapes might have similar functional features for photosynthesis, since the leaf areas can be described by the same equation. The anisotropy expressed as a difference in leaf shape for some plants might be an adaptive response to enable them to adapt to different habitats.

Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called ‘biogeom’ that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree‐rings, and three growth‐rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. ‘biogeom’ includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth‐rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. ‘biogeom’ can quantify the intra‐ or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.

The leaf area, as an important leaf functional trait, is thought to be related to leaf length and width. Our recent study showed that the Montgomery equation, which assumes that leaf area is proportional to the product of leaf length and width, applied to different leaf shapes, and the coefficient of proportionality (namely the Montgomery parameter) range from 1/2 to π/4. However, no relevant geometrical evidence has previously been provided to support the above findings. Here, four types of representative leaf shapes (the elliptical, sectorial, linear, and triangular shapes) were studied. We derived the range of the estimate of the Montgomery parameter for every type. For the elliptical and triangular leaf shapes, the estimates are π/4 and 1/2, respectively; for the linear leaf shape, especially for the plants of Poaceae that can be described by the simplified Gielis equation, the estimate ranges from 0.6795 to π/4; for the sectorial leaf shape, the estimate ranges from 1/2 to π/4. The estimates based on the observations of actual leaves support the above theoretical results. The results obtained here show that the coefficient of proportionality of leaf area versus the product of leaf length and width only varies in a small range, maintaining the allometric relationship for leaf area and thereby suggesting that the proportional relationship between leaf area and the product of leaf length and width broadly remains stable during leaf evolution.

Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.