Deepest nodes in marked ordered trees

TL;DR

This paper investigates marked ordered trees, a variation of ordered trees used to model skew Dyck paths, focusing on the number of deepest nodes and deriving explicit generating functions with asymptotic analysis.

Contribution

It introduces explicit generating functions for marked ordered trees and analyzes the average number of deepest nodes, revealing a limit of 5/3 for large trees, differing from standard ordered trees.

Findings

Average number of deepest nodes approaches 5/3 in large marked ordered trees.

Explicit generating functions for these trees are derived.

Comparison with standard ordered trees shows a different asymptotic behavior.

Abstract

A variation of ordered trees, where each rightmost edge might be marked or not, if it does not lead to an endnode, is investigated. These marked ordered trees were introduced by E. Deutsch et al.\ to model skew Dyck paths. We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches $\frac53$ when the number of nodes gets large. This is to be compared to standard ordered trees where the average number of deepest nodes approaches $2$.