# Exact and approximate limit behaviour of the Yule tree's cophenetic index

## TL;DR

This paper investigates the limit distribution of the cophenetic index in Yule trees, providing theoretical results, approximations, and practical methods for hypothesis testing of phylogenetic trees.

## Contribution

It establishes the almost sure convergence of the normalized cophenetic index and introduces heuristic simulation methods for the limit distribution.

## Key findings

- Normalized index converges almost surely and in L2.
- Limit distribution is similar to Quicksort distribution in certain cases.
- Heuristic simulation methods produce reasonable tail approximations.

## Abstract

In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure-birth tree conditioned on $n$ contemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and in L2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction-type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure-birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.