  Let $K$ be an algebraic function field of one variable with constant field
$k$ and let $C$ be the Dedekind domain consisting of all those elements of $K$
which are integral outside a fixed place $\infty$ of $K$. When $k$ is finite
the group $GL_2(C)$ plays a central role in the theory of Drinfeld modular
curves analagous to that played by $SL_2(Z)$ in the classical theory of modular
forms. When $k$ is finite (resp. infinite) we refer to a group $GL_2(C)$ as an
arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with
$GL_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the group is derived
from that of the quotient graph $GL_2(C)\backslash T$. Using an elementary
approach which refers explicitly to matrices we determine the structure of all
the vertex stabilizers of $T$. This extends results of Serre, Takahashi and the
authors. We also determine all possible valencies of the vertices of
$GL_2(C)\backslash T$ for the important special case where $\infty$ has degree
1.
