Semple, CharlesSteel, M.2016-08-142016-08-1419991172-8531http://hdl.handle.net/10092/12576Let X be a finite set and let d be a function from X x X into an arbitrary group Q. An example of such a function arises by taking a tree T whose vertices include X, assigning two elements of Q to each edge of T ( one for each orientation of the edge), and setting d(i,j) equal to the product of the elements along the directed path from i to j. We characterize conditions when an arbitrary function d can be represented in this way, and show how such a representation may be explicitly constructed. We also describe the extent to which the underlying tree and the edge weightings are unique in such a representation. These results generalize a recent theorem involving undirected edge assignments by an Abelian group. The non-Abelian bi-directed case is of particular relevance to phylogeny reconstruction in molecular biology.enAll Rights ReservedDiscussion / Working PapersFields of Research::49 - Mathematical sciences::4901 - Applied mathematics::490102 - Biological mathematics