# Maximum Likelihood Estimation for Unrooted 3-Leaf Trees: An Analytic Solution for the CFN Model

## TL;DR

This paper provides a closed-form solution for maximum likelihood estimation of unrooted 3-leaf phylogenetic trees under the CFN model.

## Contribution

A novel closed-form solution for maximum likelihood estimation in 3-leaf unrooted trees under the CFN model.

## Key findings

- A closed-form solution for maximum likelihood estimation is derived for unrooted 3-leaf trees under the CFN model.
- The paper characterizes all ways a maximum likelihood estimate can fail to exist for generic data.
- Theoretical validation is provided for predictions made in previous studies using the CFN model.

## Abstract

Maximum likelihood estimation is among the most widely-used methods for inferring phylogenetic trees from sequence data. This paper solves the problem of computing solutions to the maximum likelihood problem for 3-leaf trees under the 2-state symmetric mutation model (CFN model). Our main result is a closed-form solution to the maximum likelihood problem for unrooted 3-leaf trees, given generic data; this result characterizes all of the ways that a maximum likelihood estimate can fail to exist for generic data and provides theoretical validation for predictions made in Parks and Goldman (Syst Biol 63(5):798–811, 2014). Our proof makes use of both classical tools for studying group-based phylogenetic models such as Hadamard conjugation and reparameterization in terms of Fourier coordinates, as well as more recent results concerning the semi-algebraic constraints of the CFN model. To be able to put these into practice, we also give a complete characterization to test genericity.