Phylogenetic Diversity Over an Abelian Group

Abstract

There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G—simply add the elements on the edges that connect the leaves in that subset. This map has been well-studied in the case where G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups. We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups are also described.