The Formation of a Tree Leaf

In this article, we build a mathematical model to understand the formation of a plant leaf. Our model is based on the idea that a leaf tends to maximize internal efficiency by developing an efficient transport system for water and nutrients. From a single cell, a leaf with various cost functionals on transport systems will grow into leaves with various shapes and various venation patterns. 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 [13][14][15] 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. Different parameters will provide tree leaves with different shapes and different venation patterns. Unlike other models, the shapes of tree leaves are not predetermined here. Also, the aggregation of cells here is not random. It depends on the relative transportation cost from the cells to the root. Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf.