  We provide a proof of Sholander's claim (Trees, lattices, order, and
betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) 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 (Tree representations of betweenness relations
defined by intersection and inclusion, Mathematics and Social Sciences 185,
5-36 (2009)) of tree betweenness and provide a relatively short proof.
