# Fringe Analysis of Plane Trees Related to Cutting and Pruning

## TL;DR

This paper analyzes how rooted plane trees are affected by four different fringe reduction operations, providing exact and asymptotic results for node survival, including limit theorems.

## Contribution

It introduces a generalized framework for tree pruning operations and derives precise asymptotic formulas for node counts after repeated reductions.

## Key findings

- Exact expressions for expected surviving nodes
- Asymptotic expansions for variance
- Central limit theorems for node counts

## Abstract

Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.