Networks with side branching in biology.

There are many examples of branching networks in biology. Examples include the structure of plants and trees as well as cardiovascular and bronchial systems. In many cases these networks are self-similar and exhibit fractal scaling. In this paper we introduce the Tokunaga taxonomy for the side branching of networks and his parameterization of self-similar side-branching. We introduce several examples of deterministic branching networks and show that constructions with the same fractal dimension can have different side-branching parameters. As an example of stochastic-branching in biology we consider the vein structure of a leaf. We show that the vein structure of the leaf and river networks have nearly identical side branching statistics. We introduce diffusion limited aggregation (DLA) clusters. These clusters also exhibit Tokunaga side-branching statistics. We consider several alternative explanations for why leaves, river networks, and DLA clusters have similar side-branching statistics. We also consider the allometric scaling relation between metabolic rate and the mass of species in terms of a cardiovascular system with Tokunaga statistics. We find reasonably good agreement between the observed scaling exponent, approximately 0.75, and our model for a range of values of the fractal dimension of the network and the blood flow resistance parameter.