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The design issues affecting a parallel implementation of the alpha-beta search algorithm are discussed with emphasis on a tree decomposition scheme that is intended for use on well ordered trees. In particular, the principal variation splitting method has been implemented, and experimental results are presented which show how such refinements as progressive deepening, narrow window searching, and the use of memory tables affect the performance of multiprocessor based chess playing programs. When dealing with parallel processing systems, communication delays are perhaps the greatest source of lost time. Therefore, an implementation of our tree decomposition based algorithm is presented, one that operates with a modest amount of message passing within a network of processors. Since our system has low search overhead, the principal basis for comparison is the communication overhead, which in turn is shown to have two components.