Three or four levels of hierarchy minimize hydraulic power in leaves with pinnate dendritic venation.

We present a model of the hydraulic power required by the network of veins in a leaf with pinnate venation. The pinnate networks we study are dendritic networks with a single midrib and L levels of hierarchy, where L=2 corresponds to secondary veins branching from the midrib, L=3 additionally has tertiary veins branching from secondary veins, L=4 additionally has quaternary veins branching from tertiary veins, etc.We begin by utilizing the classic results of Murray to show that the minimal power required in a pipe of constant radius depends only on the length of the pipe and the volume flow rate to the 2/3 power. After then showing that the power required by the midrib is essentially independent of the number of secondary and higher order veins, we provide an explicit formula for the minimal total power required with L levels of hierarchy. The critical parameters in this model are the height and width of the leaf and the density ρ of vein terminations. We show how ρ can be estimated from published data on leaf vein density, and that for a very wide range of ρ the value of L minimizing the total power required is either 3 or 4. That is, three or four levels of leaf vein hierarchy suffice to minimize the required hydraulic power.