Safe and economic operation of networked systems is often challenging. Optimization-based schemes are frequently considered, since they achieve near-optimality while ensuring safety via the explicit consideration of constraints. In applications, these schemes, however, often require solving large-scale optimization problems. Iterative techniques from distributed optimization are frequently proposed for complexity reduction. Yet, they achieve feasibility only asymptotically, which induces a substantial computational burden. This work presents an approximate dynamic programming scheme, which is guaranteed to deliver a feasible solution in "one shot", i.e., in one backward-forward iteration over all subproblems provided they are coupled by a tree structure. Our proposed scheme generalizes methods from seemingly disconnected domains such as power systems and optimal control. We demonstrate its efficacy for problems with nonconvex constraints via numerical examples from both domains.