Ranked tree shapes and ranked genealogies are binary tree structures commonly used in biological areas. These trees are used to model the ancestral history of a sample, typically a sample of DNA or RNA sequences. We will discuss two representations of ranked tree shapes as constrained matrices of integers and ordered matchings. We exploit these representations to define an ergodic Markov chain on the space of ranked tree shapes with uniform stationary distribution. We will study its mixing time and compare to other related work.
This is based on joint work with Mackenzie Simper.