Computable categoricity of trees of finite height

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ3-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆n+1-categorical but not ∆n-categorical. ∗The first author was partially supported by NSF grant DMS-9732526 and by the Vilas Foundation of the University of Wisconsin. The second author was partially supported by a VIGRE grant to the University of Wisconsin. The third author was partially supported by a VIGRE postdoc under NSF grant number 9983660 to Cornell University. The fourth author was partially supported by an NSF postdoctoral fellowship.