Phylogenetic trees are fundamental tools for studies in Evolutionary Biology, Population Genetics, and Comparative Genomics. However, the algebraic and topological properties of spaces of phylogenetic trees are, by and large, unexplored. The majority of contemporary works are built under the infrastructure of BHV tree space, in which each phylogenetic tree is represented as a point and the branch lengths as its coordinates. Despite the fundamental role of the BHV space in phylogenetic inference, computational complexity of the geodesic metric for the BHV space significantly limits its applications for studying algebraic and topological properties of tree spaces. In this dissertation we propose a novel mathematical framework for phylogenetic inference and demonstrate its applications in statistical inference of phylogenetic trees.

This thesis includes two major parts. First, we develop a topological vector space in which the topology of a phylogenetic tree is defined as a linear map. We further map phylogenetic trees with branch lengths to spaces of graphical-path vectors. We show that there exists an isomorphism between phylogenetic trees and a tropical variety in Euclidean space by this vectorization mapping. In addition, the topological vector space can be metricized by the L2 norm.

In the second part, statistical properties of phylogenetic trees are studied in the context of the corresponding metric space. Based on the vectorization of trees, we define a centroid and variability measure for phylogenetic tree and propose an estimation method for the mean tree. The estimator is inferred algebraically and demonstrated by a simulation study as asymptotically normal distributed.