We introduce a low-complexity distributed slotted MAC protocol that can support all feasible arrival rates in a wireless backhaul network (WBN). For arbitrary wireless networks, such a maximum throughput protocol has been notoriously hard to realize because (i) even if global topology information is available, the problem of computing the optimal link transmission set at each slot is NP-complete (ii) no bounds exist on the number of steps required for such a computation (per-slot overhead). For the logical tree structures induced by the WBN traffic matrices, we first introduce a centralized algorithm that solves the optimal scheduling problem in a number of steps at most linear in the number of nodes in the network. This is achieved by discovering and exploiting a novel set of graph-theoretical properties of the WBN contention graph. Guided by the centralized algorithm, we design a distributed protocol where, at the beginning of each slot, nodes coordinate and incrementally compute the optimal link transmission set. We then introduce an algorithm to compute the minimum number of steps to complete this computation, thus minimizing the per-slot overhead. Using both analysis and simulations, we show that in practice our protocol yields low overhead when implemented over existing wireless technologies and significantly outperforms existing suboptimal distributed slotted scheduling mechanisms.