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Many-to-many multicast can be widely used in computer and communication networks supporting various continuous multi-media applications. This paper assumes each user is both a source and a receiver as well as a terminal in many-to-many multicasting. Under this assumption a many-to-many multicast tree appears as a terminal Steiner tree (TeST). This paper uses a heuristic idea of under a fixed TeST topology to compute a good many-to-many multicast tree. Based on this novel idea two basic problems are proposed, the minimum cost/delay many-to-many multicast tree under a fixed TeST topology problem (MCMP/MDMP). Further both of them are distributed into two types, the centralized and decentralized. This paper presents a polynomial-time exact algorithm for the centralized and decentralized MCMP/MDMP respectively.