# The variance of the average depth of a pure birth process converges to 7

## TL;DR

This paper proves that the variance of the average leaf depth in a pure birth process converges to 7, showing consistency within individual trees despite fluctuations across the ensemble.

## Contribution

It establishes that the variance of the average leaf depth in a pure birth process converges to a constant, contrasting with the linear growth of variance in individual leaf depths.

## Key findings

- Variance of average leaf depth converges to 7.
- Within individual trees, average depth is highly consistent.
- Variance across trees fluctuates but stabilizes for the average depth.

## Abstract

If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly in time. In this letter, we instead consider the variance of the average depth of leaves within each individual tree, establishing that, in contrast, it converges to a constant, $7$. This result indicates that while the variance in leaf depths amongst the ensemble of pure birth processes undergoes large fluctuations, the average depth across individual trees is much more consistent.