HORTON LAW IN SELF-SIMILAR TREES

Abstract

Self-similarity of random trees is related to the operation of pruning. Pruning [Formula: see text] 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 [Formula: see text] and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at [Formula: see text]. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers [Formula: see text] and [Formula: see text] are defined as the expected number of branches of order [Formula: see text], and the expected number of order-[Formula: see text] branches that merged order-[Formula: see text] branches, [Formula: see text], respectively, in a finite tree of order [Formula: see text]. The Tokunaga coefficients are defined as [Formula: see text]. 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: [Formula: see text]. We show that for self-similar trees, the condition [Formula: see text] is necessary and sufficient for the existence of the strong Horton law: [Formula: see text], as [Formula: see text] for some [Formula: see text] and every [Formula: see text]. 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.