Horton Law in Self-Similar Trees

Abstract. Self-similarity of random trees is related to the operation of pruning. Pruning R cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex v and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at v. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers NkrKs and NijrKs are defined as the expected number of branches of order k, and the expected number of order-i branches that merged order-j branches, j ą i, respectively, in a finite tree of order K. The Tokunaga coefficients are defined as TijrKs “ NijrKs{NjrKs. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: Tk :“ Ti,i`krKs. We show that for self-similar trees, the condition lim supkÑ8 T 1{k k ă 8 is necessary and sufficient for the existence of the strong Horton law: NkrKs{N1rKs Ñ R1 ́k, as K Ñ 8 for some R ą 0 and every k ě 1. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.