The Splits in the Neighborhood of a Tree

A phylogenetic tree represents historical evolutionary relationships between different species or organisms. The space of possible phylogenetic trees is both complex and exponentially large. Here we study combinatorial features of neighbourhoods within this space, with respect to four standard tree metrics. We focus on the splits of a tree: the bipartitions induced by removing a single edge from the tree. We characterize those splits appearing in trees that are within a given distance of the original tree, demonstrating close connections between these splits, the Whitney number of a tree, and the binary characters with a given parsimony length.