Density and fractal property of the class of oriented trees

Abstract

We show a density theorem for the class of finite proper trees ordered by the homomorphism order, where a proper tree is an oriented tree which is not homomorphic to a path. We also show that every interval of proper trees, in addition to being dense, is in fact universal. We end by considering the fractal property in the class of all finite digraphs. This complements the characterization of finite dualities of finite digraphs.