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 Σ03-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 δ0n+1-categorical but not δ0n-categorical.
"Computable categoricity of trees of finite height." J. Symbolic Logic 70 (1) 151 - 215, March 2005. https://doi.org/10.2178/jsl/1107298515