Branching Structures in Nature and Human Societies

doi:10.5047/forma.2016.s002 expressions of human relations and communication networks. Some of these examples appear in this review article. As a mathematical framework developed since many years we have the method developed by a geologist Horton (1945), who found so-called Horton’s law in the tree-type river branching structures, which is introduced briefly in Sec. 2. This method has a great advantage in a sense that the geometrical properties of tree-type structures can be expressed in terms a single parameter. On the other hand, the network-type branching systems have been often treated successfully by scientists from various fields, but they were not based on a simple method similar to that of Horton. The present author has once proposed a method to treat network-type system for leaf veins and road systems, which was included in a monograph by the present author (Takaki, 1978) and not published as a scientific paper. It is introduced in Sec. 4. Of course, the framework of analysis of branching systems is not limited to the Horton’s method, and some remarkable examples are introduced in the following sections. In particular, an application of the topology (one of mathematical fields) is made by a pathological scientist Shimizu (1992) for analysis of 3-dimensional (3D) network-type structures of blood vessels in human liver along with a topological concept called “Betti number”. Here, it is expected that introductions of various method and concepts would give a larger scope of branching systems, which would stimulate further development of studies in future.