  We study the geodesic complexity of the ordered and unordered configuration
spaces of graphs in both the $\ell_1$ and $\ell_2$ metrics. We determine the
geodesic complexity of the ordered two-point $\varepsilon$-configuration space
of any star graph in both the $\ell_1$ and $\ell_2$ metrics and of the
unordered two-point configuration space of any tree in the $\ell_1$ metric, by
finding explicit geodesics from any pair to any other pair, and arranging them
into a minimal number of continuously-varying families. In each case the
geodesic complexity matches the known value of the topological complexity.
