Subtree distances, tight spans and diversities

We characterize when a set of distances $d(x,y)$ between elements in a set X have a subtree representation, a real tree T and a collection ${\left\{S_x\right\}}_{x\in X}$ of subtrees of T such that $d(x,y)$ equals the length of the shortest path in T from a point in $S_x$ to a point in $S_y$ for all $x,y\in X$. The characterization was first established for finite X by Hirai (2006) using a tight span construction defined for distance spaces, metric spaces without the triangle inequality. To extend Hirai's result beyond finite X we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity – a generalization of a metric space which assigns values to all finite subsets of X, not just to pairs – has a tight span which is tree-like.