Markov embedding of ranked unlabelled evolutionary trees and its applications

TL;DR

This paper introduces a Markov chain embedding for ranked unlabelled trees, enabling efficient computation of tree statistics and neutrality tests in evolutionary models.

Contribution

It presents a novel Markov chain embedding that reduces state space complexity and allows for efficient analysis of tree balance and neutrality testing.

Findings

Efficient algorithms for computing Fréchet means of trees.

Joint distribution of tree balance indices derived using phase-type theory.

New neutrality tests with improved power demonstrated on simulated data.

Abstract

Rooted bifurcating trees are mathematical objects used to model evolutionary relationships and arise naturally in both coalescent theory and phylogenetics. Recent numerical representations of tree topologies, known as F-matrices, allow for summarizing a sample of trees via Fr\'echet means and provide new measures of tree balance. However, the number of ranked unlabelled trees grows super-exponentially with the number of leaves. This makes computation intensive and current methods rely on mixed integer programming and simulation-based methods. Moreover, F-matrices are difficult to interpret, and their distribution is only described in terms of first- and second-order moments under neutral branching. In this paper, we introduce a Markov chain embedding of ranked and unlabelled trees that drastically decreases the size of the state space. Leveraging this embedding, we develop an algorithm that efficiently computes all Fr\'echet means and use discrete phase-type theory to obtain the joint distribution of tree balance indices. We also use discrete phase-type theory to generalize previous results regarding moments of F-matrices to arbitrary order for any time homogeneous and bifurcating coalescent model. Using this framework, we construct three tests for neutrality and demonstrate their improved power compared to previous methods on simulated data.