Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. The spatial geometry of the phylogenetic tree is necessary to adequately quantify this uncertainty during statistical analyzes of collections of evolutionary trees that may have been inferred from biological data. Recently, the Wald space was introduced. This is the length space of trees that are a particular subset of the manifold of symmetric positive definite matrices. In this work, Wald space is formally introduced and its topology and structure are studied in detail. In particular, we show that the Wald space has an open cubic disjointly coupled topology that is contractible, and by carefully characterizing the cubic boundaries, we find that the Wald space is of type (A) Whitney stratification Indicates that it is a space. By imposing a metric induced by an affine invariant metric on a symmetric positive definite matrix, we prove that the Wald space is a geodesic Riemannian stratified space. New numerical methods are proposed and investigated for constructing geodesics, computing Fr\’echet means, and computing Wald space curvature. This work is intended to serve as a mathematical basis for further geometric and statistical studies on this space.

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