A refinement for ordered labeled trees

TL;DR

This paper introduces a new refinement of ordered labeled trees based on the size of their maximal decreasing subtree, providing a detailed classification within the set of all such trees.

Contribution

It defines and studies the set of ordered labeled trees with a specified size of maximal decreasing subtree, offering a novel structural perspective.

Findings

Characterization of trees with a given maximal decreasing subtree size

Enumeration formulas for the refined sets

Insights into the structure of ordered labeled trees

Abstract

Let $\mathcal{O}_n$ be the set of ordered labeled trees on ${0,...,n}$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study a new refinement $\mathcal{O}_{n,k}$ of $\mathcal{O}_n$, which is the set of ordered labeled trees whose maximal decreasing subtree has $k+1$ vertices.