Many plant species produce ovate leaves, but there is no general parametric model for describing this shape. Here, we used two empirical nonlinear equations, the beta and Lobry–Rosso–Flandrois (LRF) equations, and their modified forms (referred to as the Mbeta and MLRF equations for convenience), to generate bilaterally symmetrical curves along the x-axis to form ovate leaf shapes. In order to evaluate which of these four equations best describes the ovate leaf shape, we used 14 leaves from 7 Neocinnamomum species (Lauraceae) and 72 leaves from Chimonanthus praecox (Calycanthaceae). Using the AIC and adjusted root mean square error to compare the fitted results, the modified equations fitted the leaf shapes better than the unmodified equations. However, the MLRF equation provided the best overall fit. As the parameters of the MLRF equation represent leaf length, maximum leaf width, and the distance from leaf apex to the point associated with the maximum leaf width along the leaf length axis, these findings are potentially valuable for studying the influence of environmental factors on leaf shape, differences in leaf shape among closely related plant species with ovate leaf shapes, and the extent to which leaves are bilaterally symmetrical. This is the first work in which temperature-dependent developmental equations to describe the ovate leaf shape have been employed, as previous studies lacked similar leaf shape models. In addition, prior work seldom attempted to describe real ovate leaf shapes. Our work bridges the gap between theoretical leaf shape models and empirical leaf shape indices that cannot predict leaf shape profiles.

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.

Stomatal pore area and density determine the capacity for gas exchange between the leaf interior and the atmosphere. Stomatal area is given by the profile formed by two guard cells, and the cumulative stomatal area characterizes the area of leaf surface occupied by stomata. The areas of all stomata captured in a micrograph are sorted in ascending order to form a sequence, which is referred to as a sequence of stomatal area here. In total, 360 leaves of 12 Magnoliaceae species with 30 leaves for each species were sampled. For each leaf, two 662 μm × 444 μm fields of view (micrographs) of stomata were captured on the right leaf width axis. In each micrograph, the length and width of each stoma were measured, and the area of the stoma was determined using the product of stomatal length and width multiplied by a proportionality coefficient. Stomatal area sequences of Magnoliaceae in the constant field of view were found to follow a geometric series (GS). Prior studies estimated the common ratio of the GS as the mean of the quotients of any two adjacent terms, and estimated the first term as the mean of the first terms (i.e., the smallest stomatal area) represented by the quotient of each term and the estimated common ratio to a power of the order of the term minus 1, which is referred to as Method-1. However, it produced large prediction errors for some stomatal area sequences. In the present study, the nonlinear regression was used to fit the stomatal area sequences using the common ratio and the first term as two model parameters (Method-2). We compared the two methods using the mean absolute percent error (MAPE, ≤5% considered as a good fit) values of the 720 stomatal micrographs from the 12 Magnoliaceae species. The goodness of fit of Method-2 was better than that of Method-1 (52.4% MAPE values were ≤5% for Method-1 and 99.6% for Method-2). There were significant variations in the estimated common ratios, as well as the estimated first terms and the MAPE values across the 12 Magnoliaceae species, but overall, the interspecific differences in the MAPE values were small. We conclude that the GS hypothesis for the stomatal area sequences of the 12 Magnoliaceae species was further strengthened by the new method. This method further provides a valuable approach for the calculation of total stomatal area per unit leaf area.