A representation of exchangeable hierarchies by sampling from random real trees

A hierarchy on a set S, also called a total partition of S, is a collection H$$\mathcal {H}$$ of subsets of S such that S∈H$$S \in \mathcal {H}$$, each singleton subset of S belongs to H$$\mathcal {H}$$, and if A,B∈H$$A, B \in \mathcal {H}$$ then A∩B$$A \cap B$$ equals either A or B or ∅$$\varnothing $$. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H$$\mathcal {H}$$ associated as …