On the neighbourhoods of trees

Phylogenetic trees are a commonly used tool for representing the relationships between species in an evolutionary system, especially in evolutionary biology. A central task in the study of these trees is to determine which among a set of hypothesised trees gives the best explanation of empirical data. However, finding the trees that optimize some criterion is often computationally prohibitive because of the large number of trees to be checked. An approach that avoids this is a heuristic hill-climbing algorithm that searches tree space using tree rearrangement operations [5, 6]. That is, at each iteration the optimal tree within one rearrangement operation is chosen as the input for the next step, and the algorithm is thus guaranteed to find a local optimum. Loosely speaking, a tree rearrangement operation breaks a tree into two contiguous parts, and rejoins these parts to form a new tree. Among the three tree rearrangement operations of interest, namely nni (nearest neighbour interchange), spr (subtree prune and regraft) and tbr (tree bisection and reconnection), each induces a distinct metric on the space of unrooted trees. Several properties of these metrics are