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Horton-Strahler numbers for binary butterfly trees: exact analysis

Helmut Prodinger

TL;DR

This work analyzes Horton-Strahler numbers (register function) for binary butterfly trees, a binary-tree variant formed by glueing two binary trees. It develops generating-function machinery, deriving explicit closed forms for the relevant functions $R_p(z)$, $S_p(z)$, and their butterfly analogue $T_p(z)$ using the substitution $z=\frac{u}{(1+u)^2}$. By solving recurrences for $T_p(z)$ and applying Mellin-transform techniques, the paper shows that the average Horton-Strahler number for butterfly trees has the same leading logarithmic growth as for classical binary trees, up to small periodic fluctuations, with a remainder bounded by $O\big(\frac{\log^* n}{n}\big)$. These results establish exact and asymptotic parity with ordinary binary trees under the butterfly operation, enhancing understanding of how Horton-Strahler statistics behave under tree-gluing constructions.

Abstract

Peca suggested in a recent paper on the arxiv to consider binary butterfly trees and their Horton-Strahler numbers. The trees are obtained by glueing two binary trees together in a special way; the results are again binary trees but with a different probability distribution. A thorough combinatorial analysis is provided and leads asymptotically to the same results as for classical binary trees.

Horton-Strahler numbers for binary butterfly trees: exact analysis

TL;DR

This work analyzes Horton-Strahler numbers (register function) for binary butterfly trees, a binary-tree variant formed by glueing two binary trees. It develops generating-function machinery, deriving explicit closed forms for the relevant functions , , and their butterfly analogue using the substitution . By solving recurrences for and applying Mellin-transform techniques, the paper shows that the average Horton-Strahler number for butterfly trees has the same leading logarithmic growth as for classical binary trees, up to small periodic fluctuations, with a remainder bounded by . These results establish exact and asymptotic parity with ordinary binary trees under the butterfly operation, enhancing understanding of how Horton-Strahler statistics behave under tree-gluing constructions.

Abstract

Peca suggested in a recent paper on the arxiv to consider binary butterfly trees and their Horton-Strahler numbers. The trees are obtained by glueing two binary trees together in a special way; the results are again binary trees but with a different probability distribution. A thorough combinatorial analysis is provided and leads asymptotically to the same results as for classical binary trees.
Paper Structure (3 sections, 2 theorems, 33 equations, 2 figures)

This paper contains 3 sections, 2 theorems, 33 equations, 2 figures.

Table of Contents

  1. Binary trees and Horton-Strahler numbers
  2. Horton-Strahler numbers resulting from glueing together two binary trees
  3. The average value of Horton-Strahler numbers in marked binary trees

Key Result

Theorem 1

The generating function $T_p(z)$ of trees in $\mathscr{A}$ with Horton-Strahler number $\ge p$ is for $p\ge0$ given by

Figures (2)

  • Figure 1: Two binary trees $t_1$, $t_2$, the rightmost leaf of $t_1$ indicated, and then $t_2$ glued there, with resulting binary tree $t_1\oplus t_2$
  • Figure 2: The resulting tree $t_1\oplus t_2$ can originate in two possible ways from glueing two trees together.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2