Distribution and Asymptotic Behavior of the Phylogenetic Transfer Distance

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[Formula: see text] in a reference tree [Formula: see text] 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 [Formula: see text]. By taking the minimum TD from [Formula: see text] to all branches in T we define the transfer index, denoted by [Formula: see text], measuring the degree of agreement of T with [Formula: see text]. Let us consider a reference branch [Formula: see text] having p tips on its light side and define the transfer support (TS) as [Formula: see text]. 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 [Formula: see text]. Moreover, we fully characterize the distribution of [Formula: see text] 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.