Asymptotic Classes of Trees and ${\mathrm{\aleph }}_0$-categoricity

This paper focuses on the characterization of ${\mathrm{\aleph }}_0$-categorical theories of trees in the following sense: for any ${\mathrm{\aleph }}_0$-cateorical theory $T$ of trees there is a tree plan $\mathrm{\Gamma }$ such that $T=Th(\mathrm{\Gamma }(\omega ))$ where $\mathrm{\Gamma }(\omega )$ is the generic model of a Fraïssé class $\mathbf{K}(\mathrm{\Gamma })$ obtained from the tree plan $\mathrm{\Gamma }$. Also, it is shown that $\mathbf{K}(\mathrm{\Gamma })$ forms an asymptotic class, and its model-theoretic properties have been studied. Moreover, it is demonstrated that asymptotic classes of finite trees yield ${\mathrm{\aleph }}_0$-categorical ultraproducts, and a characterization of supersimple, finite rank trees, using the notion of tree plan, is provided.