A note on the relaxation time of two Markov chains on rooted phylogenetic tree spaces

Phylogenetic trees are commonly used to model the evolutionary relationships among a collection of biological species. Over the past fifteen years, the convergence properties for Markov chains defined on phylogenetic trees have been studied, yielding results about the time required for such chains to converge to their stationary distributions. In this work we derive an upper bound on the relaxation time of two Markov chains on rooted binary trees: one defined by nearest neighbor interchanges (NNI) and the other defined by subtree prune and regraft (SPR) moves.