# Balanced vertices in labeled rooted trees

## TL;DR

This paper investigates the properties of balanced vertices in labeled rooted trees, providing counts and probabilistic analyses across different tree types, including decreasing binary trees, where the likelihood of a vertex being balanced decreases with tree size.

## Contribution

It introduces a formal definition of balanced vertices in rooted trees and analyzes their distribution and probability in various tree classes, including decreasing binary trees.

## Key findings

- Probability of a random vertex being balanced decreases with tree size in decreasing binary trees.
- Provides enumeration methods for counting balanced vertices in different tree varieties.
- Establishes monotonicity of balanced vertex probability in decreasing binary trees.

## Abstract

In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.