On the enumeration of rooted trees with fixed size of maximal decreasing trees

TL;DR

This paper investigates the enumeration of rooted labeled trees based on the size of their maximal decreasing subtrees, providing new combinatorial insights into their structure.

Contribution

It introduces a refined classification of rooted labeled trees according to the size of their maximal decreasing subtrees and derives enumeration results for these classes.

Findings

Derived formulas for counting trees with a fixed size of maximal decreasing subtree

Established combinatorial properties of the refined tree classes

Provided enumeration results for various parameters of rooted trees

Abstract

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