the original Muller–Schupp theorem imply that a finitely generated group is virtually free if and only if it has context-free word problem (which is the modern formulation of the Muller–Schupp theorem). A 1983 paper of Linnell established accessibility of finitely generated groups where the orders of finite subgroups are bounded. It was later observed (see ) that Linnell's result together with the original Muller–Schupp theorem were sufficient to derive the modern statement of the Muller–Schupp theorem, without having to use Dunwoody's result. In the case of torsion-free groups, the situation is simplified as the accessibility results are not needed and one instead uses Grushko theorem about the rank of a free product. In this setting, as noted in the original Muller and Schupp paper, the Muller–Schupp theorem says that a finitely generated torsion-free group has context-free word problem if and only if this group is free. In a subsequent related paper, Muller and Schupp proved that a ``finitely generated" graph Γ has finitely many end isomorphism types if and only if Γ is the transition graph of a push-down automaton. As a consequence, they show that the monadic theory of a ``context-free" graph (such as the Cayley graph of a virtually free group) is decidable, generalizing a classic result of Rabin for binary trees. Later Kuske and Lohrey proved that virtually free groups are the only finitely generated groups whose Cayley graphs have decidable monadic theory. Bridson and Gilman applied the Muller–Schupp theorem to show that a finitely generated group admits a ``broom-like" combing if and only if that group is virtually free. Sénizergues used the Muller–Schupp theorem to show that the isomorphism problem for finitely generated virtually free group is primitive recursive. Gilman, Hermiller, Holt and Rees used the Muller–Schupp theorem to prove that a finitely generated group G is virtually free if and only if there exist a finite generating set X for G and a finite set of length-reducing rewrite rules over X whose application reduces any word to a geodesic word. Ceccherini-Silberstein and Woess consider the setting of a finitely generated group G with a finite generating set X, and a subgroup K of G such that the set of all words over the alphabet representing elements of H is a context-free language. Generalizing the setting of the Muller–Schupp theorem, Brough studied groups with poly-context-free word problem, that is where the word problem is the intersection