## Center for Applied Mathematics Colloquium

Data generated in such areas as evolutionary biology and medical imaging are frequently tree-shaped, and thus non-Euclidean in nature. As a result, standard techniques for analyzing data in Euclidean spaces become inappropriate, and new methods must be used. One such framework is the space of metric trees constructed by Billera, Holmes, and Vogtmann. This space is a non-positively curved, or CAT(0), polyhedral cone complex, with a unique geodesic (shortest path) between any two trees and a well-defined notion of a mean tree. I will present experiments demonstrating that this Frechet mean is comparable to existing summary methods, and that the Frechet variance is a more precise measure than those commonly used. I will also discuss Central Limit Theorems on this treespace. This talk is a combination of joint work with Dan Brown, and Huiling Le and Dennis Barden.