  We use the theory of arithmetic quotients of the Bruhat-Tits tree developed
by Serre and others to obtain Dirichlet-style theorems for Diophantine
approximation on global function fields. This approach allows us to find sharp
values for the constants involved and, occasionally, explicit examples of badly
approximable quadratic irrationals. Additionally, we can use this method to
easily compute the measure of the set of elements that can be written as the
limit of a sequence of ``better than expected'' approximants. All these results
can be easily obtained via continued fractions when they are available, so that
quotient graphs can be seen as a partial replacement of them when this fails to
be the case.
