Entropy rates for Horton self-similar trees

TL;DR

This paper derives an exact formula for the number of planted binary trees with specific Horton-Strahler orders and analyzes their structural complexity using entropy, including entropy rates for trees following Horton Law.

Contribution

It provides a new exact enumeration formula and analyzes the entropy rate of planted binary trees, linking structural complexity with Horton Law.

Findings

Exact formula for planted binary trees with given Horton-Strahler orders

Entropy rate quantification of structural complexity as trees grow

Analysis of trees satisfying Horton Law with Horton exponent R

Abstract

In this paper we examine planted binary plane trees. First, we provide an exact formula for the number of planted binary trees with given Horton-Strahler orders. Then, using the notion of entropy, we examine the structural complexity of random planted binary trees with N vertices. Finally, we quantify the complexity of the tree's structural properties as tree grows in size, by evaluating the entropy rate for planted binary plane trees with N vertices and for planted binary plane trees that satisfy Horton Law with Horton exponent R.