Correct. A bijection implies isomorphism in the category of sets. Thus an isomorphism exists. I suppose we need to convince ourselves that algorithms and abstract syntax trees do indeed form sets. (Exclude such things as "the algorithm that computes the set of all algorithms", etc.)
But that is typically not how the word is used because to substitute bijection with isomorphism, it would only make sense when talking about cardinality - that is not how you used it.
