THE COMPUTABLE DIMENSION OF I-TREES OF INFINITE HEIGHT

branching model, effectively infinite computable dimension. We study computable trees with distinguished initial subtree (briefly, I-trees). It is proved that all I-trees of infinite height are computably categorical, and moreover, they all have effectively infinite computable dimension. In a finite language, a model A is computable if its domain is a computable subset of ω, and its basic operations and relations are all computable. In computable model theory, algorithmic properties of algebraic systems are treated up to computable isomorphism. The number of distinct (up to computable isomorphism) computable presentations of a model A is called the computable dimension of A. If this dimension is 1 then we say that A is computably categorical. The computable categoricity of trees was studied in [1, 2]. In [1], it was proved that all computable trees of infinite height have computable dimension ω. For computable trees of finite height, in [2], it was shown that their computable dimension may assume only the value 1 or ω, and a complete characterization of computable categoricity was given. In the present paper, we study the question about spectrum of possible computable dimensions of trees enriched by an initial subtree (briefly, I-trees). It is proved that the computable dimension of an