Finite Sholander Trees , Trees , and their Betweennesses

We provide a proof of a claim made by Sholander (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees. Furthermore, we strengthen Burigana’s axiomatic characterization of so-called betweennesses induced by trees (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) and provide a short proof.