Enumerations of vertices among all rooted ordered trees with levels and degrees

TL;DR

This paper provides enumeration formulas and bijections for specific vertex types in rooted ordered trees, unifying and extending previous combinatorial results in tree enumeration.

Contribution

It introduces new enumeration formulas and bijections for various vertex classes in rooted ordered trees, generalizing prior work.

Findings

Enumerates vertices with specific degree and level conditions

Provides bijections between different vertex classes

Unifies multiple previous enumeration results

Abstract

In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\ell$, (ii) non-first-children of degree $k$ at level $\ell-1$, (iii) leaves having $k-1$ elder siblings at level $\ell$, and (iv) non-leaves of outdegree $k$ at level $\ell-1$. Our results unite and generalize several previous works in the literature.