Enumerations of vertices among all rooted ordered trees with levels and degrees
Sen-Peng Eu, Seunghyun Seo, Heesung Shin

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 at level , (ii) non-first-children of degree at level , (iii) leaves having elder siblings at level , and (iv) non-leaves of outdegree at level . Our results unite and generalize several previous works in the literature.
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