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.