Weighted unary-binary trees, Hex-trees, marked ordered trees, and related structures

TL;DR

This paper explores various specialized tree structures, including Hex-trees and marked ordered trees, analyzing their properties, Horton-Strahler numbers, and asymptotic behaviors, with connections to known integer sequences.

Contribution

It introduces explicit results and asymptotic evaluations for Hex-trees and related structures, expanding understanding of their combinatorial properties.

Findings

Explicit formulas for Horton-Strahler numbers of Hex-trees

Asymptotic evaluations of tree structures

Connections to sequence A002212 in integer sequences

Abstract

Hex-trees are identified as a particular instance of weighted unary-binary trees. The Horton-Strahler numbers of these objects are revisited, and, thanks to a substitution that is not immediately intuitive, explicit results are possible. They are augmented by asymptotic evaluations as well. Furthermore, marked ordered trees (in bijection to skew Dyck paths) are investigated, followed 3-Motzkin paths and multi-edge trees. The underlying theme is sequence A002212 in the Encyclopedia of integer sequences.