Distributions of topological tree metrics between a species tree and a gene tree

In order to conduct a statistical analysis on a given set of phylogenetic gene trees, we often use a distance measure between two trees. In a statistical distance-based method to analyze discordance between gene trees, it is a key to decide "biological meaningful" and "statistically well-distributed" distance between trees. Thus, in this paper, we study the distributions of the three tree distance metrics: the edge difference, the path difference, and the precise $K$ interval cospeciation distance, between two trees: first, we focus on distributions of the three tree distances between two random unrooted trees with $n$ leaves ($n \geq 4$); and then we focus on the distributions the three tree distances between a fixed rooted species tree with $n$ leaves and a random gene tree with $n$ leaves generated under the coalescent process with given the species tree. We show some theoretical results as well as simulation study on these distributions.