  Let K be a function field with constant field k and let "infinity" be a fixed
place of K. Let C be the Dedekind domain consisting of all those elements of K
which are integral outside "infinity". The group G=GL_2(C) is important for a
number of reasons. For example, when k is finite, it plays a central role in
the theory of Drinfeld modular curves. Many properties follow from the action
of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows
how a presentation for G can be derived from the structure of the quotient
graph (or fundamental domain) G\T. The shape of this quotient graph (for any G)
is described in a fundamental result of Serre. However there are very few known
examples for which a detailed description of G\T is known. (One such is the
rational case, C=k[t], i.e. when K has genus zero and "infinity" has degree
one.) In this paper we give a precise description of G\T for the case where the
genus of K is zero, K has no places of degree one and "infinity" has degree
two. Among the known examples a new feature here is the appearance of vertex
stabilizer subgroups (of G) which are of quaternionic type.
