An order-theoretic tree is a partial order such that the set of nodes larger that any node is linearly ordered. Rooted trees, ordered by the ancestor relation, are order-theoretic trees. The ternary relation B(x, y, z) of betweenness states that y is between x and z, in some sense. For linear and partial orders, for finite and order-theoretic trees, it has been characterized in first-order or monadic second-order logic. In an order-theoretic tree, B(x, y, z) means that x < y < z or z < y < x or x < y ≤ x ⊔ z or z < y ≤ x ⊔ z where x ⊔ z is the least upper-bound of incomparable x and z. We prove that the class of the induced substructures of the betweenness structures of countable order-theoretic trees is monadic second-order definable. The proof uses partitioned probe cographs, a variant of cographs, and their known five bounds, i.e., the five minimal excluded induced subgraphs, up to isomorphism. This article links two apparently unrelated topics: cographs and order-theoretic trees.