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Distributed optimization problems comprise a number of subsystems, each with their own objective and local constraints, together with coupling conditions between the subsystems. If these coupling conditions are linear and the overall objective consists of the sum of the subsystem objectives, the overall problem becomes partially separable. Such systems are called tree-sparse or tree-coupled if, in addition, the coupling between subsystems is only enforced along an underlying tree graph. Prominent examples of this class of problems stem from scenario tree approaches for robust optimization. Standard solution methods typically lead to a sequence of highly structured linear subproblems of symmetric saddle-point type. We will show how to design efficient preconditioners for their solution using Krylov subspace methods.