With the proliferation of available data in the biological field of Phylogenetics, certain computational methods become obsolete and there is a need for more algorithms that can take advantage of both the increased quality and quantity of data while also optimizing computational complexity. In particular, the problem of averaging a set of phylogenetic trees has been an outstanding problem due to the fact that it is difficult to find a biologically meaningful metric that takes into account both a tree’s morphology and it’s respective branch lengths. In a 2001 paper, Billera et. al. solved this problem by introducing what they call tree space, a topological space in which every phylogenetic tree can be represented as a point. This thesis provides the necessary undergraduate-level background in Graph Theory, Combinatorics, and Topology required to construct this space. The remainder of the thesis consists of characterizing the shortest path between two phylogenetic trees in this space as well as defining an algorithm for computed the average of a finite set of trees. This algorithm has important applications in Phylogenetics (and any other discipline with data in the form of trees) because it is the first algorithm that computes the tree average while taking into account both tree morphology and branch lengths. In addition, recent papers have reduced the computational complexity of the algorithm to polynomial time
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