The problem of scheduling time-critical messages over a tree network is considered. Messages arrive at any of the nodes and have to reach the root node before their deadlines expire, else they are considered lost, The network is assumed to be operating in discrete time and the messages need one time unit for transmission from one node to the next along its path, The arrival and deadline processes are arbitrary, The policy which transmits messages with smallest extinction (arrival + deadline) time at every link is shown to minimize the number of lost messages over all time intervals and for every sample path.