Let $A$ be a free group and $G = A*_t$. When is $G$ also a free group? Suppose $t y t^{1} = z$ and there is a splitting $A = B*C$ so that $y \in B$ and $z \in C$ and $z$ is a member of some basis of $C$ then clearly $G$ is free. Is this the only case that $G$ is free?
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Yes. This is a theorem of Shenitzer. For a modern treatment see, for instance, this recent paper of Louder. I give a proof of a similar fact in section 2 of this preprint. 

