  We address the problem of strategyproof (SP) facility location mechanisms on
discrete trees. Our main result is a full characterization of onto and SP
mechanisms. In particular, we prove that when a single agent significantly
affects the outcome, the trajectory of the facility is almost contained in the
trajectory of the agent, and both move in the same direction along the common
edges. We show tight relations of our characterization to previous results on
discrete lines and on continuous trees. We then derive further implications of
the main result for infinite discrete lines.
