The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains

TL;DR

This paper explores the mathematical structure of multifurcating genealogical trees, providing enumeration, lattice structure, and Markov chain methods to improve inference of complex ancestral histories.

Contribution

It introduces a comprehensive framework for analyzing multifurcating trees, including enumeration, lattice construction, and Markov chain models, filling a gap in coalescent inference research.

Findings

Enumerated the space of multifurcating ranked trees.

Constructed a lattice structure on the tree space.

Derived bounds for Markov chain mixing times.

Abstract

Coalescent models of bifurcating genealogies are used to infer evolutionary parameters from molecular data. However, there are many situations where bifurcating genealogies do not accurately reflect the true underlying ancestral history of samples, and a multifurcating genealogy is required. The space of multifurcating genealogical trees, where nodes can have more than two descendants, is largely underexplored in the setting of coalescent inference. In this paper, we examine the space of rooted, ranked, and unlabeled multifurcating trees. We recursively enumerate the space and then construct a partial ordering which induces a lattice on the space of multifurcating ranked tree shapes. The lattice structure lends itself naturally to defining Markov chains that permit exploration on the space of multifurcating ranked tree shapes. Finally, we prove theoretical bounds for the mixing time of two Markov chains defined on the lattice, and we present simulation results comparing the distribution of trees and tree statistics under various coalescent models to the uniform distribution on this tree space.