Trees and streams: The efficiency of branching patterns

Extending the analysis of branching patterns of the drainage net of rivers, originated by Horton, the relation of average numbers and lengths of tree branches to size of branch was investigated. Size of branch was defined by branch order, or its position in the hierarchy of tributaries. It was found that, as in river drainage nets, there is a definite logarithmic relation between branch order and lengths and numbers. This definite relation is quantitatively comparable, within limits, among river networks, tree branching systems, and several random-walk models in both two and three dimensions. Such a relation appears to be the most probable under the applicable constraints. Moreover the most probable arrangement appears to minimize the total length of all stems in the branching system within other constraints and so, to that extent, achieves a certain efficiency.