# Expected size of a tree in the fixed point forest

## TL;DR

This paper investigates the local limit of the fixed-point forest, an infinite random tree derived from a permutation sorting algorithm, and computes the expected size and leaves of its subtrees.

## Contribution

It generalizes the fixed-point forest model and provides explicit calculations for the expected size, leaves, and variance bounds of subtrees within this model.

## Key findings

- Expected size of a subtree is computed.
- Expected number of leaves in a subtree is derived.
- Bounds on the variance of subtree sizes are established.

## Abstract

We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process configuration on $[0,1]^\mathbb{N}$. We generalize this random tree, and compute the expected size and expected number of leaves of a random rooted subtree in the generalized version. We also obtain bounds on the variance of the size.