Refined enumeration of vertices among all rooted ordered $d$-trees

TL;DR

This paper provides a comprehensive enumeration of vertices with certain outdegree and level constraints in rooted ordered d-trees, unifying and extending previous enumeration results.

Contribution

It introduces a generalized enumeration framework for vertices in rooted ordered d-trees, broadening the scope of prior specific cases.

Findings

Derived formulas for vertex counts with outdegree ≥ k and level ≥ ℓ

Unified multiple previous enumeration results into a single framework

Extended enumeration techniques to more general d-tree structures

Abstract

In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.