  We study the first-order almost-sure theories for classes of finite
structures that are specified by homomorphically forbidding a set $\mathcal{F}$
of finite structures. If $\mathcal{F}$ consists of undirected graphs, a full
description of these theories can be derived from the
Kolaitis-Pr\"omel-Rothschild theorem, which treats the special case where
$\mathcal{F} = \{K_n\}$. The corresponding question for finite sets
$\mathcal{F}$ of finite directed graphs is wide open. We present a full
description of the almost-sure theories of classes described by homomorphically
forbidding finite sets $\mathcal{F}$ of oriented trees; all of them are
$\omega$-categorical. In our proof, we establish a result of independent
interest, namely that every constraint satisfaction problem for a finite
digraph has first-order convergence, and that the corresponding asymptotic
theory can be described as a finite linear combination of $\omega$-categorical
theories.
