  Relatively prime pairs of integers can be represented as nodes in three way
branching trees. We construct trees of B\'ezout coefficients which correspond
to the relatively prime pairs in the aforementioned trees. As one application,
we compare the B\'ezout coefficients in these trees to those returned by the
gcd function in Matlab. As another application, we use these trees to decrease
the computation time required to create computer generated hyperbolic wallpaper
designs.
