The relation between hierarchical and euclidean models for psychological distances

In one well-known model for psychological distances, objects such as stimuli are placed in a hierarchy of clusters like a phylogenetic tree; in another common model, objects are represented as points in a multidimensional Euclidean space. These models are shown theoretically to be mutually exclusive and exhaustive in the following sense. The distances among a set ofn objects will be strictly monotonically related either to the distances in a hierarchical clustering system, or else to the distances in a Euclidean space of less thann — 1 dimensions, but not to both. Consequently, a lower bound on the number of Euclidean dimensions necessary to represent a set of objects is one less than the size of the largest subset of objects whose distances satisfy the ultrametric inequality, which characterizes the hierarchical model.