Journal of Symbolic Logic 70 (1):151-215 (2005)

Abstract
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
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DOI 10.2178/jsl/1107298515
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References found in this work BETA

Generic Copies of Countable Structures.Chris Ash, Julia Knight, Mark Manasse & Theodore Slaman - 1989 - Annals of Pure and Applied Logic 42 (3):195-205.
Categoricity in Hyperarithmetical Degrees.C. J. Ash - 1987 - Annals of Pure and Applied Logic 34 (1):1-14.
On the Complexity of Categoricity in Computable Structures.Walker M. White - 2003 - Mathematical Logic Quarterly 49 (6):603.

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D-Computable Categoricity for Algebraic Fields.Russell Miller - 2009 - Journal of Symbolic Logic 74 (4):1325 - 1351.
Computable Dimension for Ordered Fields.Oscar Levin - 2016 - Archive for Mathematical Logic 55 (3-4):519-534.

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