  We investigate intersections of geodesic lines in $H^2$ and in an associated
tree T, proving the following result. Let M be a punctured hyperbolic torus and
let $\gamma$ be a closed geodesic in M. Any edge of any triangle formed by
distinct geodesic lines in the preimage of $\gamma$ in $H^2$ is shorter then
$\gamma$. However, a similar result does not hold in the tree T. Let W be a
reduced and cyclically reduced word in $\pi_1(M) = <x,y>$. We construct several
examples of triangles in T formed by distinct axes in T stabilized by
conjugates of W such that an edge in those triangles is longer than L(W). We
also prove that if W overlaps two of its conjugates in such a way that the
overlaps cover all of W and the overlaps do not intersect, then there exists a
decomposition $W = BC^kI; k > 0$, with B a terminal subword of C and I an
initial subword of C.
