Distribution and asymptotic behavior of the phylogenetic transfer distance

Abstract : The transfer distance (TD) was introduced in the classification framework and studied in the context of phylogenetic tree matching. Recently, Lemoine et al. (Nature 556(7702):452-456, 2018. https://doi.org/10.1038/s41586-018-0043-0) showed that TD can be a powerful tool to assess the branch support on large phylogenies, thus providing a relevant alternative to Felsenstein's bootstrap. This distance allows a reference branch β in a reference tree T to be compared to a branch b from another tree T (typically a bootstrap tree), both on the same set of n taxa. The TD between these branches is the number of taxa that must be transferred from one side of b to the other in order to obtain β. By taking the minimum TD from β to all branches in T we define the transfer index, denoted by φ(β, T), measuring the degree of agreement of T with β. Let us consider a reference branch β having p tips on its light side and define the transfer support (TS) as 1 − φ(β, T)/(p − 1). Lemoine et al. (2018) used computer simulations to show that the TS defined in this manner is close to 0 for random "bootstrap" trees. In this paper, we demonstrate that result mathematically: when T is randomly drawn, TS converges in probability to 0 when n tends to ∞. Moreover, we fully characterize the distribution of φ(β, T) on caterpillar trees, indicating that the convergence is fast, and that even when n is small, moderate levels of branch support cannot appear by chance.