Extended Similarity Trees

Trees are commonly used to represent proximity relations that emerge, for instance, from studies of classification, similarity, and identification. Trees are employed to describe the data, explore their structure and model their generating process. They offer a convenient graphical display that is readily interpretable in terms of a hierarchy of clusters (Sokal &Sneath, 1963) or in terms of common and distinctive features (Tversky, 1977). The simplest tree structure is the hierarchical clustering model (Jardine & Sibson, 1971; Johnson, 1967) based on the ultrametric inequality, which states that for any triple of points the two larger distances are equal. That is, any three points can be labeled x, y, z such that d(x, z) = d(y, z) >_ d(x, y). This assumption gives rise to a tree in which all the endpoints (leaves) are equally distant from the root. The ultrametric tree is highly restrictive because any two elements of one cluster must be equally similar to any other element outside the cluster. This restriction is relaxed in the additive tree (e.g., Cunningham, 1978; Sattath & Tversky, 1977), where the leaves are not necessarily equidistant from the root. The additive tree provides greater flexibility than the ultrametric tree, but it too cannot accomodate (nonnested) overlapping clusters because any two clusters in a tree are either nested or disjoint. Throughout the paper we use the standard abbreviations (e.g., HICLUS, ADDTREE, ADCLUS) for scaling algorithms, and the unabbreviated forms (e.g., hierarchical clustering, additive tree, additive clustering) for the respective models. This article describes a new representation of proximity relations, called an extended tree, which accommodates nonnested feature structures while maintaining the basic property of a tree that every pair of points is joined by a unique path. To motivate and