The space of metric phylogenetic trees, as constructed by Billera, Holmes, and Vogtmann, is a polyhedral cone complex. This space is non-positively curved, which ensures there is a unique shortest path (geodesic) between any two trees, and that the mean and variance of a set or distribution of trees is well-defined. Furthermore, there is a polynomial time algorithm to compute geodesics, which leads to a practical algorithm for computing mean trees. I will present some applications of this mean and variance to some biological problems, such as constructing species trees from gene trees and understanding the effect of sequence length on tree reconstruction. This is joint work with Ezra Miller and Scott Provan.