In this paper we present an extensive study of dynamic routing on trees under the “matching with consumption” routing model. We present an asymptotically optimal on-line algorithm which routes k packets to their destination within d(k−1) + d · dist routing steps, where d is the degree of tree T on which the routing takes place and dist is the maximum distance any packet has to travel. We also present an off-line algorithm that solves the same problem within 2(k−1) + dist steps. The analysis of our algorithms is based on the establishment of a close relationship between the matching and the hot-potato routing models.