On individual leaf depths of trees

TL;DR

This paper introduces a generating function approach to analyze the distribution of individual node depths and related statistics in various combinatorial structures, providing exact and asymptotic results.

Contribution

It extends previous methods to track individual distributions in trees and paths, offering new exact formulas and asymptotic insights.

Findings

Derived exact formulas for node depth distributions in trees

Provided asymptotic results refining known formulas

Established combinatorial proofs for probabilistic results

Abstract

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature.