Physical Limits to Leaf Size in Tall Trees

Leaf sizes in angiosperm trees vary by more than 3 orders of magnitude, from a few mm to over 1 m. This large morphological freedom is, however, only expressed in small trees, and the observed leaf size range declines with tree height, forming well-defined upper and lower boundaries. The vascular system of tall trees that distributes the products of photosynthesis connects distal parts of the plant and forms one of the largest known continuous microfluidic distribution networks. In biological systems, intrinsic properties of vascular systems are known to constrain the morphological freedom of the organism. We show that the limits to leaf size can be understood by physical constraints imposed by intrinsic properties of the carbohydrate transport network. The lower boundary is set by a minimum energy flux, and the upper boundary is set by a diminishing gain in transport efficiency.

Figure 1

Examples of leaf size variability. Typical leaf lamina length $l$ (indicated by arrows) for a group of trees of height $h\le 30\mathrm{m}$ growing in North America. Leaf size varies from 3 cm in Ulmus parvifolia (Lacebark elm) to 60 cm in Magnolia macrophylla (Bigleaf magnolia). For compound leaves (left), leaflet length was used (dashed arrow). See the Supplemental Material Fig. 1 [8] for a complete list of species in the figure.

Figure 2

Variation in leaf size $l$ with tree height $h$ based on botanical data covering 1925 species from 327 genera and 93 families; see Supplemental Material Table I [8]. Gray triangles show the reported range of leaf sizes for particular species as the longest (▵) and shortest (▽) leaf lamina length plotted as a function of tree height $h$. Circles show the five longest (red, dark gray) and five shortest (green, light gray) leaves in each 20 m height bin for trees taller than $h=20\mathrm{m}$. Solid lines are fits to Eq. (2) (upper limit) and Eq. (3) (lower limit) with parameters corresponding to a minimum flow speed of $u_{\min }\simeq 100\mu \mathrm{m}/\mathrm{s}$ and energy output efficiency of $\sim 90\%$; see Figs. 3a, 3b and text for details. Dashed lines indicate 95% confidence intervals.

Figure 3

Limits to leaf size in tall trees. (a) Mechanism for the upper limit on leaf size. Energy flux $E$ plotted as a function of leaf size $l$ (thin black line). As the leaf becomes longer, the energy flux slowly approaches a maximum value $E_{\max }=kc{r}^2\Delta p/(8\eta h)$ set by the height of the tree (thick black line). In the case of an $h=50\mathrm{m}$ tree, the gain in net energy export is very small beyond $l=40\mathrm{cm}$, where the energy flux is 90% of $E_{\max }$ (dashed line) and the leaf length must double to $l=80\mathrm{cm}$ to obtain a 5% increase in net energy flux $E$. This leads to the prediction of maximum leaf size $l_{\max }$ given in Eq. (2). (b) Mechanism for lower limit on leaf size. Energy flux $E$ plotted as a function of leaf size $l$ (thin black line). As the leaf length increases, the energy flux increases above a minimum required energy flux $E_{\min }=kc{u}_{\min }$ (thick black line), which is assumed to be independent of tree height. This leads to the prediction that the leaf size $l_{\min }$ at which $E=E_{\min }$ is given by Eq. (3). Here, the case of a $h=50\mathrm{m}$ tree, where $l_{\min }\simeq 2\mathrm{cm}$, is shown. In (a) and (b), gray leaves marked with (*) have the same size.