We study a combinatorial problem that is moti- vated by "client-server" schedulers for networks of workstations. Using a number of FIFO queues the scheduler is required to schedule N-leaf binary (or any fixed degree) trees in such a way that each non- leaf node of the tree being scheduled is executed before its children. We establish a tradeoff between the number of which we view as queues used by the algorithm- measuring the control complexity of the algorithm - and the memory requirements of the algorithm, as embodied in the required capacity of the largest- capacity queue. Let 2(N) denote the minimax per-queue capac- ity for a k-queue algorithm that schedules all N-leaf binary trees, and let 2(N) denote the analogous quantity for complete binary trees. We establish the following bounds. For all N, k≤ log N, 1 (2N-1)1/k k log N+1 ≤Q(N) ≤2N1/+1, 1 N1/k k (log N+1)1-1/k N ≤ Q(N) ≤ (4k) 1-1/k N1/k log1-1/k N The bound for complete binary trees is tight, to within constant factors, for all fixed k.