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

Abstract

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