  We define train track maps for graphs-of-groups $\cal G$ and exhibit the
precise conditions under which the fundamental finiteness properties known for
classical train track maps extend to this generalization. These finiteness
properties are the crucial tool to control the decrease of illegal turns under
iteration of the train track map, and they are a principal ingredient in the
answer to basic algorithmic questions about automorphisms induced by such train
track maps on $\pi_1 \cal G$.
