New Results for Self-Similar Trees with Applications to River Networks

TLDR: The self-similar trees (SSTs) as mentioned in this paper are a subclass of tree graphs based on the Strahler ordering scheme, which is defined in terms of a generator matrix which acts as a "blueprint" for constructing different trees.

Abstract: In a little-known series of papers beginning in 1966, Tokunaga introduced an infinite class of tree graphs based on the Strahler ordering scheme. As recognized by Tokunaga (1984), these trees are characterized by a self-similarity property, so we will refer to them as self-similar trees, or SSTs. SSTs are defined in terms of a generator matrix which acts as a “blueprint” for constructing different trees. Many familiar tree constructions are absorbed as special cases. However, in Tokunaga's work an additional assumption is imposed which restricts from SSTs to a much smaller class. We will refer to this subclass as Tokunaga's trees. This paper presents several new and unifying results for SSTs. In particular, the conditions under which SSTs have well-defined Horton-Strahler stream ratios are given, as well as a general method for computing these ratios. It is also shown that the diameters of SSTs grow like mβ, where m is the number of leaves. In contrast to many other tree constructions, here β need not equal 1/2; thus SSTs offer an explanation for Hack's law. Finally, it is demonstrated that large discrepancies exist between the predictions of Shreve's well-known model and detailed measurements for large river networks, while other SSTs fit the data quite well. Other potential applications of the SST framework include diffusion-limited aggregation (DLA), lightning, bronchial passages, neural networks, and botanical trees.