In hot-potato (deflection) routing, nodes in the network have no buffers for packets in transit, so that some conflicting packets must be deflected away from their destinations. In this work, we study one-tomany batch routing problems on arbitrary tree topologies with n nodes. The routing time of a routing algorithm is the time for the last packet to reach its destination. Denote by rt∗ the optimal routing time for a given routing problem. We construct the first hot-potato routing algorithms whose routing times are asymptotically nearoptimal; that is, the incurred routing times are within polylogarithmic factors from optimal. More specifically, we present: