A matroid associated with a phylogenetic tree

A (pseudo-)metric D on a finite set X is said to be a ‘tree metric’ if there is a finite tree with leaf set X and nonnegative edge weights so that, for all x, y ∈ X , D(x, y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is – up to canonical isomorphism – uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree’s edge weights in this case. Thus, it seems of some interest to investigate which subsets of ( X 2 )