Let $ C $ be the set of simple closed geodesics on a once-punctured torus
with any complete finite-area hyperbolic metric.
McShane's identity says that
$ \sum_{c \in C} {1 \over 1+e^{l(c)}} = {1 \over 2} $
where $ l(c) $ denotes the hyperbolic length of the closed geodesic $ c $.
We translate this into a statement about trees of Markoff triples,
and hence give a new proof of this result.
Bull. London. Math. Soc. 28 (1996) 73-78.