# Scaling of semi-permeable membrane with leaf size

**Geometry**Level 3

From here out, we'll consider the flow of water from one vein of one leaf, through the trunk, to the site of consumption, as a proxy for the overall vascular network\(^1\).

Over the course of the evolution of leaved plants, nature has figured out efficient vein patterns to facilitate sugar transport from the various parts of the leaf to the stem.

Suppose that the geometric layout shown in the figure above is the pattern found for all trees, and that it holds for all leaves, regardless of their size. The average distance, \(\langle d \rangle\) traveled by some parcel of sap as it travels to the base of the leaf scales as some function of leaf length \(l\)

\[\displaystyle \langle d \rangle \sim f(l)\sim l^{\alpha}\]

What is \(\alpha\)?

\(^1\)Convince yourself this is a legitimate reduction of the problem of sucrose transport in the cell.