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. We discuss some unexpected properties of the mean and convex hulls in this space, and give a polynomial time algorithm for computing convex hulls in the space of trees with 5 leaves. This algorithm extends to any 2D CAT(0) polyhedral complex with a single vertex.