# Protection numbers in simply generated trees and P\'olya trees

## TL;DR

This paper analyzes the asymptotic behavior of protection numbers in various classes of trees, providing exact formulas and efficient numerical methods based on generating functions and singularity analysis.

## Contribution

It offers new asymptotic results and explicit formulas for the expectation and variance of protection numbers in multiple tree models.

## Key findings

- Expected protection number converges to a limit as tree size grows.
- Derived exact sum formulas for protection number statistics.
- Provided efficient numerical methods for high-accuracy calculations.

## Abstract

We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P\'olya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging and therefore allowing an efficient numerical computation of high accuracy. Most proofs are based on a singularity analysis of generating functions.