This paper presents a prescriptive account of diagnostic problem solving, or diagnosis, in quality and process control. This paper identifies a general strategy, named branch-and-prune, whose manifestations can be found in disciplines such as medical diagnosis, troubleshooting of devices, and model-based diagnosis in artificial intelligence. The work aims to offer a clear conceptualization of this strategy, based on the notions of structures for the search space, and constraints to the cause's nature. The idea is to treat the search space of candidate explanations as a tree structure, in which general and high-level causal directions are branched into more specific and detailed explanations. Constraints eliminate all but a few branches (pruning), which are explored in more detail. We enumerate eight generic structures as a basis for branching the search tree. We demonstrate that our conceptualization in terms of structures and constraints gives a rationale for generally known methods and heuristics in quality engineering and operations management. The paper contributes a unifying conceptual understanding of a class of diagnostic techniques, and it improves the strategy's operationality by offering generic structures, and a simpler and more flexible account of its working. A description of a real-life quality problem solving effort forms a tangible basis for the discussion.