A note on plane trees with decreasing labels

TL;DR

This paper investigates the asymptotic bounds for counting labeled plane trees with decreasing labels along root-to-leaf paths and explores an application to eigenvalue calculation of the adjacency matrix.

Contribution

It provides new asymptotic bounds for the number of such labeled trees and demonstrates an application to spectral graph theory.

Findings

Derived asymptotic upper and lower bounds for the count of labeled plane trees

Applied results to compute the largest eigenvalue of a tree's adjacency matrix

Enhanced understanding of label restrictions in combinatorial tree structures

Abstract

This note derives asymptotic upper and lower bounds for the number of planted plane trees on $n$ nodes assigned labels from the set $\{1,2,\ldots, k\}$ with the restriction that on any path from the root to a leaf, the labels must strictly decrease. We illustrate an application to calculating the largest eigenvalue of the adjacency matrix of a tree.