The height of multiple edge plane trees

TL;DR

This paper studies multi-edge plane trees, establishing a bijection with classical d-ary trees for bounded out-degrees, and analyzes their height and vertex distribution asymptotically.

Contribution

It introduces a bijection between bounded out-degree multi-edge trees and classical d-ary trees, enabling new asymptotic analyses of their height and vertex counts.

Findings

Height distribution of multi-edge trees analyzed asymptotically

Number of vertices distribution characterized asymptotically

Bijection simplifies analysis by relating to classical d-ary trees

Abstract

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.