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) GT. 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 GT 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 GT 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.
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