# Growing and Destroying Catalan-Stanley Trees

## TL;DR

This paper studies a deterministic growth process for Catalan-Stanley trees, a special class of rooted plane trees linked to Dyck paths with odd-length returns, analyzing their growth dynamics and age distribution.

## Contribution

It introduces a growth procedure for Catalan-Stanley trees and provides asymptotic analysis of their age and growth speed.

## Key findings

- Asymptotic distribution of tree age for large sizes
- Growth speed comparison between trees and ancestors
- Characterization of the growth process dynamics

## Abstract

Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan-Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan-Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan-Stanley tree of given size as well as for the "speed" of the growth process by comparing the size of a given tree to the size of its ancestors.