Hierarchical trees can be perfectly scaled in one dimension

Holman (1972) proved theorems which led him to suggest that there was a fundamental opposition between hierarchical clustering and non-metric Euclidean multidimensional scaling. Empirical experience has shown this to be untrue. Explanations of this apparent contradiction have been offered previously by Kruskal (1977) and Critchley (1986). In this paper we point out the feasibility of perfectly scaling a hierarchical tree in one dimension when the primary approach to ties (Kruskal 1964) is taken. Indeed, there is a whole polyhedral convex cone of solutions for which we obtain an explicit expression.