On minimum spanning tree-like metric spaces

We attempt to shed new light on the notion of ‘tree-like’ metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space M on n points, when does a fully labelled positive-weighted tree T exist on the same n vertices that precisely realises M using its shortest path metric? We prove that if a spanning tree representation, T, of M exists, then it is isomorphic to the unique minimum spanning tree in the weighted complete graph associated with M, and we introduce a fourth-point condition that is necessary and sufficient to ensure the existence of T whenever each distance in M is unique. In other words, a finite median graph, in which each geodesic distance is distinct, is simply a tree. Provided that the tie-breaking assumption holds, the fourth-point condition serves as a criterion for measuring the goodness-of-fit of the minimum spanning tree to M, i.e., the spanning tree-likeness of M. It is also possible to evaluate the spanning path-likeness of M. These quantities can be measured in O(n⁴) and O(n³) time, respectively.