  In this paper we study the action of the fundamental group of a finite metric
graph on its universal covering tree. We assume the graph is finite, connected
and the degree of each vertex is at least three. Further, we assume an
irrationality condition on the edge lengths. We obtain an asymptotic for the
number of elements in a fixed conjugacy class for which the associated
displacement of a given base vertex in the universal covering tree is at most
$T$. Under a mild extra assumption we also obtain a polynomial error term.
