An adjacent-swap Markov chain on coalescent trees

The standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper, we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has mixing time at least of order $n^3$, and at most of order $n^{4}$. Bayesian inference methods rely on Markov chain Monte Carlo methods on the space of trees. Thus, it is important to define good Markov chains which are easy to simulate and for which rates of convergence can be studied.