One central problem in AI is to compare a structural situation with another situation. Comparison makes it clear which parts of the structures are different and/or considered to be the same. Numerous methods were proposed in relation to analogical reasoning, case-based reasoning, etc. Most of them, however, lack sufficiently formalized treatment of background knowledge defining the semantics in the structures. We discuss a general framework for structural comparison, and then formulate under this framework a method of comparing concept structures by introducing the notion of coverning. It should be noted that, in a concept structure, the concepts specified by background knowledge based descriptions are partially ordered by a generic/specific relation found in background knowledge. The comprison, called here BK-comparison, of two concept structures proceeds by finding a correspondence which (1) satisfies that if the concepts in one structure have a generic/specific relation, their corresponding concepts in another structure do have the same relation, and (2) produces an MSC-covering defined as a set of highly specific generalized descriptions, each of which refers to two corresponding similar concepts in the different structures. The BK-comprison is speeded up by a new beam-search method that employs the facility of the summed levels to estimate whether or not the found coverings are the MSC-coverings, and the directional schemata guiding the search toward a successful direction. The proposed method is verified computationally by comparing concept structures induced from artificial and real-world data. Its applicability to various subject areas is also discussed.