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We study the first-order almost-sure theories for classes of finitestructures that are specified by homomorphically forbidding a set $\mathcal{F}$of finite structures. If $\mathcal{F}$ consists of undirected graphs, a fulldescription of these theories can be derived from theKolaitis-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 fulldescription of the almost-sure theories of classes described by homomorphicallyforbidding finite sets $\mathcal{F}$ of oriented trees; all of them are$\omega$-categorical. In our proof, we establish a result of independentinterest, namely that every constraint satisfaction problem for a finitedigraph has first-order convergence, and that the corresponding asymptotictheory can be described as a finite linear combination of $\omega$-categoricaltheories.

Asymptotic Theories of Classes Defined by Forbidden Homomorphisms