# Protection Number in Plane Trees

## TL;DR

This paper investigates the protection number in plane trees, analyzing its distribution for random trees and nodes using generating functions and singularity analysis, providing new insights into their probabilistic properties.

## Contribution

It introduces a detailed analysis of the protection number in random plane trees and nodes, applying singularity analysis to derive new probabilistic results.

## Key findings

- Distribution of protection number in random trees
- Protection number of a random node in a random tree
- Asymptotic behavior of protection numbers

## Abstract

The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size $n$ and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size $n$. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.