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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 any computable I-tree of infinite height is ω. Moreover, this dimension is effectively infinite, in the sense that, given any uniformly presented list of computable copies of the same I-tree, we can construct another computable copy of that tree, which is not computably isomorphic to any of the copies on the list. Notice that the results obtained can be naturally generalized to the case of several distinguished initial subtrees.

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