Abstract: Bamboo is an important component in subtropical and tropical forest communities. The plant has characteristic long lanceolate leaves with parallel venation. Prior studies have shown that the leaf shapes of this plant group can be well described by a simplified version (referred to as SGE-1) of the Gielis equation, a polar coordinate equation extended from the superellipse equation. SGE-1 with only two model parameters is less complex than the original Gielis equation with six parameters. Previous studies have seldom tested whether other simplified versions of the Gielis equation are superior to SGE-1 in fitting empirical leaf shape data. In the present study, we compared a three-parameter Gielis equation (referred to as SGE-2) with the two-parameter SGE-1 using the leaf boundary coordinate data of six bamboo species within the same genus that have representative long lanceolate leaves, with >300 leaves for each species. We sampled 2000 data points at approximately equidistant locations on the boundary of each leaf, and estimated the parameters for the two models. The root–mean–square error (RMSE) between the observed and predicted radii from the polar point to data points on the boundary of each leaf was used as a measure of the model goodness of fit, and the mean percent error between the RMSEs from fitting SGE-1 and SGE-2 was used to examine whether the introduction of an additional parameter in SGE-1 remarkably improves the model’s fitting. We found that the RMSE value of SGE-2 was always smaller than that of SGE-1. The mean percent errors among the two models ranged from 7.5% to 20% across the six species. These results indicate that SGE-2 is superior to SGE-1 and should be used in fitting leaf shapes. We argue that the results of the current study can be potentially extended to other lanceolate leaf shapes.

Abstract: Plants have diverse leaf shapes that have evolved to adapt to the environments they have experienced over their evolutionary history. Leaf shape and leaf size can greatly influence the growth rate, competitive ability, and productivity of plants. However, researchers have long struggled to decide how to properly quantify the complexity of leaf shape. Prior studies recommended the leaf roundness index (RI = 4πA/P2) or dissection index (DI = ), where P is leaf perimeter and A is leaf area. However, these two indices merely measure the extent of the deviation of leaf shape from a circle, which is usually invalid as leaves are seldom circular. In this study, we proposed a simple measure, named the ellipticalness index (EI), for quantifying the complexity of leaf shape based on the hypothesis that the shape of any oval leaf can be regarded as a variation from a standard ellipse. 2220 leaves from nine species of Magnoliaceae were sampled to check the validity of the EI. We also tested the validity of the Montgomery equation (ME), which assumes a proportional relationship between leaf area and the product of leaf length and width, because the EI actually comes from the proportionality coefficient of the ME. We also compared the ME with five other models of leaf area. The ME was found to be the best model for calculating leaf area based on consideration of the trade-off between model fit vs. complexity, which strongly supported the robustness of the EI for describing oval leaf shape. The new index can account for both leaf shape and size, and we conclude that it is a promising method for quantifying and comparing oval leaf shapes across species in future studies.

Abstract: According to Thompson’s principle of similarity, the area of an object should be proportional to its length squared. However, leaf area–length data of some plants have been demonstrated not to follow the principle of similarity. We explore the reasons why the leaf area–length allometry deviates from the principle of similarity and examine whether there is a general model describing the relationship among leaf area, width and length. We sampled more than 11,800 leaves from six classes of woody and herbaceous plants and tested the leaf area–length allometry. We compared six mathematical models based on root-mean-square error as the measure of goodness-of-fit. The best supported model described a proportional relationship between leaf area and the product of leaf width and length (i.e., the Montgomery model). We found that the extent to which the leaf area–length allometry deviates from the principle of similarity depends upon the extent of variation of the ratio of leaf width to length. Estimates of the parameter of the Montgomery model ranged between 1/2, which corresponds to a triangular leaf with leaf length as its height and leaf width as its base, and π/4, which corresponds to an elliptical leaf with leaf length as its major axis and leaf width as its minor axis, for the six classes of plants. The narrow range in practice of the Montgomery parameter implies an evolutionary stability for the leaf area of large-leaved plants despite the fact that leaf shapes of these plants are rather different.