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We provide a proof of Sholander’s claim [M. Sholander, Trees, lattices, order, and betweenness, Proceedings of the American Mathematical Society 3 (1952) 369–381] concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana’s characterization [L. Burigana, Tree representations...
For a finite poset P = (V, ≤ ), let ${\cal B}_s(P)$ consist of all triples (x,y,z) ∈ V3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let ${\cal B}_s(G)$ consist of all triples (x,y,z) ∈ V3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations ${\cal B}_s(P)$ and...
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