This talk is about the almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a finite set F of finite structures. If F consists of undirected graphs, a full description of these theories can be derived from the Kolaitis-Proemel-Rothschild theorem, which treats the special case where F = {K_n}. The corresponding question for finite sets F of directed graphs is wide open. We present a description of the almost-sure theories of classes described by homomorphically forbidding finite sets F of oriented trees; all of them are countably categorical. Joint work with Colin Jahel