Tree representations of non-symmetric group-valued proximities

Let X be a finite set and let d be a function from X X X into an arbitrary group G. An example of such a function arises by taking a tree T whose vert ices include X, assigning two elements of G 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 wh en an arbitrary function d can be represented in this way, and show how suc h a representation may be explicitly constructed. We also describe the exte nt to which the underlying tree and the edge weightings are unique in such a representation. These results generalize a recent theorem involving undir ected edge assignments by an Abelian group. The non-Abelian bi-directed cas e is of particular relevance to phylogeny reconstruction in molecular biolo gy. (C) 1999 Academic Press.