Significant correlation between leaf vein length per unit area and stomatal density: evidence from Red Tip and Chinese photinias.

The vascular veins in photosynthetic leaves play an important role in transporting water and sugars throughout the plant body, and their venation pattern and vein density determine the hydraulic efficiency of the leaf. Likewise, stomatal density (SD) can influence photosynthetic gas exchange. However, the correlation between leaf vein density and SD is seldom reported. Herein, we examined 16 leaves from the hybrid Photinia × fraseri and 16 leaves from one of its parents, P. serratifolia, to explore the correlation between leaf vein density and SD. For each leaf, equidistant lamina quadrats were excised along two longitudinal transects (one along the midrib and another along the leaf margin). For each quadrat, micrographs of 1.2 mm × 0.9 mm stomatal imprints, and 2.51 mm × 1.88 mm micrographs of leaf veins were used to measure total vein area per leaf unit area (VAA) and total vein length per unit area (VLA), as indicators of leaf vein density, to determine the correlation between SD and leaf vein density. For each taxon, there was no significant correlation between SD and VAA, but there was a significant correlation between SD and VLA. The data indicate that SD is not positively correlated with VAA but positively correlated with VLA for both the hybrid and the parent species. This study indicates that future work should focus on the relationships between SD and total vein length per unit area rather than on total leaf vein area per unit area within and across species.

Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution.

The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the original performance equation (PE-1) and its generalized counterpart (GPE-1)-to fit rotated and right-shifted Lorenz curves. Notably, another potential performance equation (PE-2), capable of generating similar curves to PE-1, has been overlooked and not systematically compared with PE-1 and GPE-1. Furthermore, PE-2 has been extended into a generalized version (GPE-2). In the present study, we conducted a comparative analysis of these four performance equations, evaluating their applicability in describing Lorenz curves related to plant organ (leaf and fruit) size. Leaf area was measured on 240 culms of dwarf bamboo (Shibataea chinensis Nakai), and fruit volume was measured on 31 field muskmelon plants (Cucumis melo L. var. agrestis Naud.). Across both datasets, the root-mean-square errors of all four performance models were consistently smaller than 0.05. Paired t-tests indicated that GPE-1 exhibited the lowest root-mean-square error and Akaike information criterion value among the four performance equations. However, PE-2 gave the best close-to-linear behavior based on relative curvature measures. This study presents a valuable tool for assessing the inequality of plant organ size distribution.

Scaling relationships among the mass of eggshell, albumen, and yolk in six precocial birds.

The proportions in the size of the avian egg albumen, yolk, and shell are crucial for understanding bird survival and reproductive success, because their relationships with volume and surface area can affect ecological and life history strategies. Prior studies have focused on the relationship between the albumen and the yolk, but little is known about the scaling relationship between eggshell mass and shape, and the mass of the albumen and the yolk. Toward this end, 691 eggs of six precocial species were examined, and their 2-D egg profiles were photographed and digitized. The explicit Preston equation, which assumes bilateral symmetrical geometry, was used to fit the 2-D egg profiles and to calculate surface areas and volumes based on the hypothesis that eggs can be treated as solids of profile revolution. The scaling relationships of eggshell mass (Ms), albumen mass (Ma), and yolk mass (My), as well as the surface area (S), volume (V), and total mass (Mt) were determined. The explicit Preston equation was validated in describing the 2-D egg profiles. The scaling exponents of Ma vs. Ms, My vs. Ms, and My vs. Ma were smaller than unity, indicating that increases in Ma and My fail to keep pace with increases in Ms, and that increases in My fail to keep pace with increases in Ma. Therefore, increases in unit nutrient contents (i.e., the yolk) involve disproportionately larger increases in eggshell mass and disproportionately larger increases in albumen mass. The data also revealed a 2/3-power scaling relationship between S and V for each species, i.e., simple Euclidean geometry is obeyed. These findings help to inform our understanding of avian egg construction and reveal evolutionary interspecific trends in the scaling of egg shape, volume, mass, and mass allocation.

Evidence That Field Muskmelon (Cucumis melo L. var. agrestis Naud.) Fruits Are Solids of Revolution.

In nature, the fruit shapes of many plants resemble avian eggs, a form extensively studied as solids of revolution. Despite this, the hypothesis that egg-shaped fruits are themselves solids of revolution remains unvalidated. To address this, 751 Cucumis melo L. var. agrestis Naud. fruits were photographed, and the two-dimensional (2D) boundary coordinates of each fruit profile were digitized. Then, the explicit Preston equation (EPE), a universal egg-shape model, was used to fit the 2D boundary coordinates to obtain the estimates of the EPE's parameters of each fruit. Under the hypothesis that egg-shaped fruits are solids of revolution, the fruit volumes were estimated using the solid of revolution formula based on the estimated EPE's parameters. To test whether the fruits are solids of revolution, the fruit volumes were measured by using a graduated cylinder and compared with the estimated volumes using the solid of revolution formula. The EPE was demonstrated to be valid in describing the 2D profiles of C. melo var. agrestis fruits. There was a significant correlation between the measured fruit volumes using the graduated cylinder and the estimated fruit volumes using the solid of revolution formula based on the estimated EPE's parameters. Acknowledging potential measurement errors, particularly fruit fuzz causing air bubbles during volume measurements, we recognize slight deviations between measured volumes and estimated values. Despite this, our findings strongly suggest that C. melo var. agrestis fruits are solids of revolution. This study contributes insights into the evolutionary aspects of fruit geometries in plants with egg-shaped fruits and introduces a practical tool for non-destructively calculating fruit volume and surface area based on photographed 2D fruit profiles.

An Inverse Scaling Relationship between Stomatal Density and Mean Nearest Neighbor Distance: Evidence from a Photinia Hybrid and One of Its Parents.

Stomata are involved in transpiration and CO2 uptake by mediating gas exchange between internal plant tissues and the atmosphere. The capacity for gas exchange depends on stomatal density (SD), stomatal size, and pore dimensions. Most published work on stomatal quantification has assumed that stomatal distribution and stomatal density are spatially homogeneous across the leaf, but this assumption has been seldom tested. We selected 32 leaves from a Photinia hybrid, Photinia × fraseri 'Red Robin', and one of its parents, P. serratifolia. For each leaf, the leaf surface was divided into three or four equidistant layers along the apical-basal axis, and, in each layer, two positions, one closer to the midrib and the other closer to the leaf margin, were further selected. We calculated SD and mean nearest neighbor distance (MNND) for each lamina section and tested the scaling relationship between SD and MNND of the sampled stomatal centers using reduced major axis protocols. In addition, we calculated the stomatal aggregation index (SAI) for each lamina section to examine the spatial arrangement of stomata at the given size of field of view of 1.2 mm × 0.9 mm. We observed that SD decreased from the lamina apex towards the base for central lamina areas but varied little at leaf margins. An inverse scaling relationship between SD and MNND was observed for both species. This relationship could be used for SD estimation using the rapidly estimated trait, MNND. SAI did not vary significantly throughout leaf lamina, and the numerical values of SAI for all fields of view were greater than one, which indicates significant spatial repulsion between stomata. The study suggests that SD varies across leaf lamina to fine-tune plant water use and maximize carbon gain. However, spatial structures of stomata from different lamina sections exhibit similar patterns (i.e., spatial inhibition between stomata at small scales), probably due to hierarchical leaf vein patterns.

Plant Morphology and Function, Geometric Morphometrics, and Modelling: Decoding the Mathematical Secrets of Plants.

Functional plant traits include a plant's phenotypic morphology, nutrient element characteristics, and physiological and biochemical features, reflecting the survival strategies of plants in response to environmental changes [...].

Inequality Measure of Leaf Area Distribution for a Drought-Tolerant Landscape Plant.

Measuring the inequality of leaf area distribution per plant (ILAD) can provide a useful tool for quantifying the influences of intra- and interspecific competition, foraging behavior of herbivores, and environmental stress on plants' above-ground architectural structures and survival strategies. Despite its importance, there has been limited research on this issue. This paper aims to fill this gap by comparing four inequality indices to measure ILAD, using indices for quantifying household income that are commonly used in economics, including the Gini index (which is based on the Lorenz curve), the coefficient of variation, the Theil index, and the mean log deviation index. We measured the area of all leaves for 240 individual plants of the species Shibataea chinensis Nakai, a drought-tolerant landscape plant found in southern China. A three-parameter performance equation was fitted to observations of the cumulative proportion of leaf area vs. the cumulative proportion of leaves per plant to calculate the Gini index for each individual specimen of S. chinensis. The performance equation was demonstrated to be valid in describing the rotated and right shifted Lorenz curve, given that >96% of root-mean-square error values were smaller than 0.004 for 240 individual plants. By examining the correlation between any of the six possible pairs of indices among the Gini index, the coefficient of variation, the Theil index, and the mean log deviation index, the data show that these indices are closely related and can be used interchangeably to quantify ILAD.

A simple way to calculate the volume and surface area of avian eggs.

Egg geometry can be described using Preston's equation, which has seldom been used to calculate egg volume (V) and surface area (S) to explore S versus V scaling relationships. Herein, we provide an explicit re-expression of Preston's equation (designated as EPE) to calculate V and S, assuming that an egg is a solid of revolution. The side (longitudinal) profiles of 2221 eggs of six avian species were digitized, and the EPE was used to describe each egg profile. The volumes of 486 eggs from two avian species predicted by the EPE were compared with those obtained using water displacement in graduated cylinders. There was no significant difference in V using the two methods, which verified the utility of the EPE and the hypothesis that eggs are solids of revolution. The data also indicated that V is proportional to the product of egg length (L) and maximum width (W) squared. A 2/3-power scaling relationship between S and V for each species was observed, that is, S is proportional to (LW2 )2/3 . These results can be extended to describe the shapes of the eggs of other species to study the evolution of avian (and perhaps reptilian) eggs.

Effects of Plant Coverage on the Abundance of Adult Mosquitos at an Urban Park.

People who take a walk in urban parks, including or adjacent to a water body such as a river, a pond, or a lake, usually suffer from mosquito bites in summer and early autumn. The insects can negatively affect the health and mood of these visitors. Prior studies about the effects of landscape composition on the abundance of mosquitos have usually taken stepwise multiple linear regression protocols to look for the landscape variables that can significantly affect the abundance of mosquitos. However, those studies have largely overlooked the nonlinear effects of landscape plants on the abundance of mosquitos. In the present study, we compared the multiple linear regression (MLR) with the generalized additive model (GAM) based on the trapped mosquito abundance data obtained by using photo-catalytic CO2-baited lamps placed at the Xuanwu Lake Park, a representative subtropical urban scenic spot. We measured the coverage of trees, shrubs, forbs, proportion of hard paving, proportion of water body, and coverage of aquatic plants within a distance of 5 m from each lamp's location. We found that MLR and GAM both detected the significant influences of the coverage of terrestrial plants on the abundance of mosquitos, but GAM provided a better fit to the observations by relaxing the limitation of the linear relationship hypothesis by MLR. The coverage of trees, shrubs, and forbs accounted for 55.2% of deviance, and the coverage of shrubs had the greatest contribution rate among the three predictors, accounting for 22.6% of the deviance. The addition of the interaction between the coverage of trees and that of shrubs largely enhanced the goodness of fit, and it increased the explained deviance of the GAM from 55.2% to 65.7%. The information in this work can be valuable for the planning and design of landscape plants to reduce the abundance of mosquitos at special urban scenic points.

Leaf-age and petiole biomass play significant roles in leaf scaling theory.

Foliage leaves are essential for plant survival and growth, and how plants allocate biomass to their leaves reveals their economic and ecological strategies. Prior studies have shown that leaf-age significantly influences leaf biomass allocation patterns. However, unravelling the effects of ontogeny on partitioning biomass remains a challenge because it is confounded by the effects of environmental factors. Here, we aim to elucidate whether leaf-age affects the allocation to the lamina and petiole by examining leaves of known age growing in the same general environmental context. We sampled 2698 Photinia serratifolia leaves developing in the same environment from April to November 2021, representing eight leaf-ages (n > 300 for each leaf-age). Petiole and lamina biomass, and lamina area were measured to evaluate the scaling relationships using reduced major axis regression protocols. The bootstrap percentile method was used to determine the differences in scaling exponents among the different leaf-ages. ANOVA with Tukey's HSD was used to compare the ratios of petiole and lamina biomass to lamina area across the leaf-ages. Correlation tests were used to determine if exponents, intercepts, and ratios differed significantly across the different leaf-ages. The data indicated that (i) the ratio of petiole and lamina biomass to lamina area and the scaling exponent of lamina biomass versus lamina area correlate positively with leaf-age, and (ii) the scaling exponent of petiole biomass versus lamina area correlates negatively with leaf-age. Leaf maturation process involves an inverse proportional allocation between lamina and petiole biomass for expanding photosynthetic area. This phenomenon underscores the effect of leaf-age on biomass allocation and the importance of adopting an ontogenetic perspective when entertaining plant scaling theories and unravelling the principles governing shifts in biomass allocation throughout the leaf lifespan.

Comparison of Two Simplified Versions of the Gielis Equation for Describing the Shape of Bamboo Leaves.

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.

Diminishing returns: A comparison between fresh mass vs. area and dry mass vs. area in deciduous species.

"Diminishing returns" in leaf economics occurs when increases in lamina mass (M), which can either be represented by lamina dry mass (DM) or fresh mass (FM), fail to produce proportional increases in leaf surface area (A), such that the scaling exponent (α) for the M vs. A scaling relationship exceeds unity (i.e., α > 1.0). Prior studies have shown that FM vs. A is better than DM vs A in assessing diminishing returns in evergreen species. However, the superiority of FM vs. A over DM vs. A has been less well examined for deciduous species. Here, we applied reduced major axis protocols to test whether FM vs. A is better than DM vs. A to describe the M vs. A scaling relationship, using a total of 4271 leaves from ten deciduous and two evergreen tree species in the Fagaceae and Ulmaceae for comparison. The significance of the difference between the scaling exponents of FM vs. A and DM vs. A was tested using the bootstrap percentile method. Further, we tested the non-linearity of the FM (DM) vs. A data on a log-log scale using ordinary least squares. We found that (i) the majority of scaling exponents of FM vs. A and DM vs. A were >1 thereby confirming diminishing returns for all 12 species, (ii) FM vs. A was more robust than DM vs. A to identify the M vs. A scaling relationship, (iii) the non-linearity of the allometric model was significant for both DM vs. A and FM vs. A., and (iv) the evergreen species of Fagaceae had significantly higher DM and FM per unit area than other deciduous species. In summary, FM vs. A is a more reliable measure than DM vs. A when dealing with diminishing returns, and deciduous species tend to invest less biomass in unit leaf light harvesting area than evergreen species.

Comparison of Leaf Shape between a Photinia Hybrid and One of Its Parents.

Leaf shape and size can vary between hybrids and their parents. However, this has seldom been quantitatively tested. Photinia × fraseri is an important landscaping plant in East Asia as a hybrid between evergreen shrubs P. glabra and P. serratifolia. Its leaf shape looks like that of P. serratifolia. To investigate leaf shape, we used a general equation for calculating the leaf area (A) of broad-leaved plants, which assumes a proportional relationship between A and product of lamina length (L) and width (W). The proportionality coefficient (which is referred to as the Montgomery parameter) serves as a quantitative indicator of leaf shape, because it reflects the proportion of leaf area A to the area of a rectangle with L and W as its side lengths. The ratio of L to W, and the ellipticalness index were also used to quantify the complexity of leaf shape for elliptical leaves. A total of >4000 leaves from P. × fraseri and P. serratifolia (with >2000 leaves for each taxon) collected on a monthly basis was used to examine: (i) whether there is a significant difference in leaf shape between the two taxa, and (ii) whether there is a monotonic or parabolic trend in leaf shape across leaf ages. There was a significant difference in leaf shape between the two taxa (p < 0.05). Although there were significant differences in leaf shape on a monthly basis, the variation in leaf shape over time was not large, i.e., leaf shape was relatively stable over time for both taxa. However, the leaf shape of the hybrid was significantly different from its parent P. serratifolia, which has wider and more elliptical leaves than the hybrid. This work demonstrates that variations in leaf shape resulting from hybridization can be rigorously quantified and compared among species and their hybrids. In addition, this work shows that leaf shape does not changes as a function of age either before or after the full expansion of the lamina.

Quantifying the Variation in the Geometries of the Outer Rims of Corolla Tubes of Vinca major L.

Many geometries of plant organs can be described by the Gielis equation, a polar coordinate equation extended from the superellipse equation, r=a|cosm4φ|n2+|1ksinm4φ|n3-1/n1. Here, r is the polar radius corresponding to the polar angle φ; m is a positive integer that determines the number of angles of the Gielis curve when φ ∈ [0 to 2π); and the rest of the symbols are parameters to be estimated. The pentagonal radial symmetry of calyxes and corolla tubes in top view is a common feature in the flowers of many eudicots. However, prior studies have not tested whether the Gielis equation can depict the shapes of corolla tubes. We sampled randomly 366 flowers of Vinca major L., among which 360 had five petals and pentagonal corolla tubes, and six had four petals and quadrangular corolla tubes. We extracted the planar coordinates of the outer rims of corolla tubes (in top view) (ORCTs), and then fitted the data with two simplified versions of the Gielis equation with k = 1 and m = 5: r=acos54φn2+sin54φn3-1/n1 (Model 1), and r=acos54φn2+sin54φn2-1/n1 (Model 2). The adjusted root mean square error (RMSEadj) was used to evaluate the goodness of fit of each model. In addition, to test whether ORCTs are radially symmetrical, we correlated the estimates of n2 and n3 in Model 1 on a log-log scale. The results validated the two simplified Gielis equations. The RMSEadj values for all corolla tubes were smaller than 0.05 for both models. The numerical values of n2 and n3 were demonstrated to be statistically equal based on the regression analysis, which suggested that the ORCTs of V. major are radially symmetrical. It suggests that Model 1 can be replaced by the simpler Model 2 for fitting the ORCT in this species. This work indicates that the pentagonal or quadrangular corolla tubes (in top view) can both be modeled by the Gielis equation and demonstrates that the pentagonal or quadrangular corolla tubes of plants tend to form radial symmetrical geometries during their development and growth.

The Modified Brière Equation and Its Applications.

The Brière equation (BE) is widely used to describe the effect of temperature on the development rate of insects, and it can produce both symmetrical and asymmetrical bell-shaped curves. Because of its elasticity in curve fitting, the integrated form of BE has been recommended for use as a sigmoid growth equation to describe the increase in plant biomass with time. However, the start time of growth predicted by the sigmoid growth equation based on the BE is not completely comparable to empirical crop growth data. In the present study, we modified the BE by adding an additional parameter to further increase its elasticity for data fitting. We termed this new equation the modified Brière equation (MBE). Data for the actual height and biomass of 15 species of plants (with two cultivars for one species) were fit with the sigmoid growth equations based on both the BE and MBE assuming that the growth start time was zero for both. The goodness of fit of the BE and MBE sigmoid growth equations were compared based on their root-mean-square errors and the corresponding absolute percentage error between them when fit to these data. For most species, we found that the MBE sigmoid growth equation achieved a better goodness of fit than the BE sigmoid growth equation. This work provides a useful tool for quantifying the ontogenetic or population growth of plants.

'biogeom': An R package for simulating and fitting natural shapes.

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.

Tree Size Influences Leaf Shape but Does Not Affect the Proportional Relationship Between Leaf Area and the Product of Length and Width.

The Montgomery equation predicts leaf area as the product of leaf length and width multiplied by a correction factor. It has been demonstrated to apply to a variety of leaf shapes. However, it is unknown whether tree size (measured as the diameter at breast height) affects leaf shape and size, or whether such variations in leaf shape can invalidate the Montgomery equation in calculating leaf area. Here, we examined 60 individual trees of the alpine oak (Quercus pannosa) in two growth patterns (trees growing from seeds vs. growing from roots), with 30 individuals for each site. Between 100 and 110 leaves from each tree were used to measure leaf dry mass, leaf area, length, and width, and to calculate the ellipticalness index, ratio of area between the two sides of the lamina, and the lamina centroid ratio. We tested whether tree size affects leaf shape, size, and leaf dry mass per unit area, and tested whether the Montgomery equation is valid for calculating leaf area of the leaves from different tree sizes. The diameters at breast height of the trees ranged from 8.6 to 96.4 cm (tree height ranged from 3 to 32 m). The diameter at breast height significantly affected leaf shape, size, and leaf dry mass per unit area. Larger trees had larger and broader leaves with lower leaf dry mass per unit area, and the lamina centroid was closer to the leaf apex than the leaf base. However, the variation in leaf size and shape did not negate the validity of the Montgomery equation. Thus, regardless of tree size, the proportional relationship between leaf area and the product of leaf length and width can be used to calculate the area of the leaves.

Comparison of a universal (but complex) model for avian egg shape with a simpler model.

Recently, a universal equation by Narushin, Romanov, and Griffin (hereafter, the NRGE) was proposed to describe the shape of avian eggs. While NRGE can simulate the shape of spherical, ellipsoidal, ovoidal, and pyriform eggs, its predictions were not tested against actual data. Here, we tested the validity of the NRGE by fitting actual data of egg shapes and compared this with the predictions of our simpler model for egg shape (hereafter, the SGE). The eggs of nine bird species were sampled for this purpose. NRGE was found to fit the empirical data of egg shape well, but it did not define the egg length axis (i.e., the rotational symmetric axis), which significantly affected the prediction accuracy. The egg length axis under the NRGE is defined as the maximum distance between two points on the scanned perimeter of the egg's shape. In contrast, the SGE fitted the empirical data better, and had a smaller root-mean-square error than the NRGE for each of the nine eggs. Based on its mathematical simplicity and goodness-of-fit, the SGE appears to be a reliable and useful model for describing egg shape.

Scaling relationships of leaf vein and areole traits versus leaf size for nine Magnoliaceae species differing in venation density.

PREMISE: Across species, main leaf vein density scales inversely with leaf area (A). Yet, minor vein density manifests no clear relationship with respect to A, despite having the potential to provide important insights into the trade-off among the investments in leaf mechanical support, hydraulics, and light interception. METHODS: To examine this phenomenon, the leaves of nine Magnoliaceae leaves were sampled, and the scaling relationships among A and midrib length (ML), total vein length (TVL), total vein area (TVA), total areole area (TAA), and mean areole area (MAA) were determined. The scaling relationships between MAA and areole density (the number of areoles per unit leaf area) and between MAA and A were also analyzed. RESULTS: For five of the nine species, A was proportional to ML2 . For eight of the nine species, TVL and TVA were both proportional to A. The numerical values of the scaling exponents for TAA vs. A were between 1.0 and 1.07 for eight species; i.e., as expected, TAA was isometrically proportional to A. There was no correlation between MAA and A, but MAA scaled inversely with respect to areole density for each species. CONCLUSIONS: The correlation between midrib "density" (i.e., ML/A) and A, and the lack of correlation between total leaf vein density and A result from the A ∝$\propto $ ML2 scaling relationship and the proportional relationship between TVL and A, respectively. Leaves with the same size can have widely varying MAA. Thus, leaf size itself does not directly constrain leaf hydraulic efficiency and redundancy.