Does the group given by this presentation have an element of order 2?

Suppose $G$ has the presentation $\langle t, x_1, x_2, ... | R \rangle$ where each relator in $R$ has the form $t^{-1}x_it = x_j$ for some $i,j$. Does $G$ have an element of order 2?

This is an HNN extension of a free group, if that changes anything.

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Note that my answer only works when $G$ is in fact an HNN-extension of a free group, which isn't obviously the case for all groups with the kind of presentation you've given. (For instance, the relations could be $t^{-1}x_i t = x_1$ for all $i$, and so $t$ isn't giving you an isomorphism between subgroups.) Could that be the source of the problem? – Richard Kent Apr 16 '11 at 4:19