The formation of a tree leaf
In this article, we build a mathematical model to understand the formation of a tree leaf. Our model is based on the idea that a leaf tends to maximize internal efficiency by developing an efficient transport system for transporting water and nutrients. The meaning of "the efficient transport system" may vary as the type of the tree leave varies. In this article, we will demonstrate that tree leaves have different shapes and venation patterns mainly because they have adopted different efficient transport systems. The efficient transport system of a tree leaf built here is a modified version of the optimal transport path, which was introduced by the author in [Comm. Cont. Math. 5 (2003) 251-279; Calc. Var. Partial Differ. Equ. 20 (2004) 283-299; Boundary regularity of optimal transport paths, Preprint] to study the phenomenon of ramifying structures in mass transportation. In the present paper, the cost functional on transport systems is controlled by two meaningful parameters. The first parameter describes the economy of scale which comes with transporting large quantities together, while the second parameter discourages the direction of outgoing veins at each node from differing much from the direction of the incoming vein. Under the same initial condition, efficient transport systems modeled by different parameters will provide tree leaves with different shapes and different venation patterns. Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf. It demonstrates that optimal transportation plays a key role in the formation of tree leaves.