Leaf-to-leaf distances in Catalan trees

TL;DR

This paper analyzes the average leaf-to-leaf distances in Catalan trees, providing explicit formulas and asymptotic analysis using generating functions, which enhances understanding of tree structures in combinatorics.

Contribution

It introduces an explicit formula for leaf-to-leaf distances in Catalan trees and employs a novel generating function approach for asymptotic analysis.

Findings

Derived an explicit formula for average leaf-to-leaf distances

Established asymptotic properties of these distances

Connected leaf-to-leaf distances to root-to-leaf path lengths

Abstract

We study the average leaf-to-leaf path lengths on ordered Catalan tree graphs with $n$ nodes and show that these are equivalent to the average length of paths starting from the root node. We give an explicit analytic formula for the average leaf-to-leaf path length as a function of separation of the leaves and study its asymptotic properties. At the heart of our method is a strategy based on an abstract graph representation of generating functions.