  Singerman introduced to the theory of maps on surfaces an object that is a
universal cover for any map. This object is a tessellation of the hyperbolic
plane together with a certain subset of the ideal boundary. The 1-skeleton of
this tessellation comprises the edges of an infinite tree whose vertices belong
to the ideal boundary. Here we show how this tree can be used to give a
beautiful geometric representation of even-integer continued fractions. We use
this representation to prove some of the fundamental theorems on even-integer
continued fractions that are already known, and we also prove some new theorems
with this technique, which have familiar counterparts in the theory of regular
continued fractions.
