Binary clustering

In many clustering systems (hierarchies, pyramids and more generally weak hierarchies) clusters are generated by two elements only. This paper is devoted to such clustering systems (called binary clustering systems). It provides some basic properties, links with (closed) weak hierarchies and some qualitative versions of bijection theorems that occur in Numerical Taxonomy. Moreover, a way to associate a binary clustering system to every clustering system is discussed. Finally, introducing the notion of weak ultrametrics, a bijection between indexed weak hierarchies and weak ultrametrics is obtained (the standard theorem involves closed weak hierarchies and quasi-ultrametrics).