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We introduce a first-order theory of finite full binary trees and thenidentify decidable and undecidable fragments of this theory. We show that theanalogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-onereduction of Post`s Correspondence Problem. By a different method, we show thatdeciding truth of sentences with one existential quantifier and one boundeduniversal quantifier is undecidable.

Undecidability in First-Order Theories of Term Algebras Extended with a Substitution Operator