# A Class of Random Recursive Tree Algorithms with Deletion

## TL;DR

This paper studies a class of random recursive trees that grow and shrink via probabilistic node addition and deletion, analyzing their size and structure across different regimes using generating functions.

## Contribution

It introduces a class of deletion rules ensuring uniform distribution of the tree conditioned on size and derives asymptotic properties of the trees.

## Key findings

- Tree size exhibits three regimes based on insertion probability p.
- Expected leaf count and root degree are asymptotically estimated.
- Results are robust across different deletion rules within the class.

## Abstract

We examine a discrete random recursive tree growth process that, at each time step, either adds or deletes a node from the tree with probability $p$ and $1-p$, respectively. Node addition follows the usual uniform attachment model. For node removal, we identify a class of deletion rules guaranteeing the current tree $T_n$ conditioned on its size is uniformly distributed over its range. By using generating function theory and singularity analysis, we obtain asymptotic estimates for the expectation and variance of the tree size of $T_n$ as well as its expected leaf count and root degree. In all cases, the behavior of such trees falls into three regimes determined by the insertion probability: $p < 1/2$, $p = 1/2$ and $p > 1/2$. Interestingly, the results are independent of the specific class member deletion rule used.