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We show polynomial time algorithms for deciding hereditary history preserving bisimilarity (in O(n^3logn)) and history preserving bisimilarity (in O(n^6)) on the class Basic Parallel Processes. The latter algorithm also decides a number of other non-interleaving behavioural equivalences (e.g., distributed bisimilarity) which are known to coincide with history preserving bisimilarity on this class. The common general scheme of both algorithms is based on a fixpoint characterization of the equivalences for tree-like labelled event structures. The technique for realizing the greatest fixpoint computation in the case of hereditary history preserving bisimilarity is based on the revealed tight relationship between equivalent tree-like labelled event structures. In the case of history preserving bisimilarity, a technique of deciding classical bisimilarity on acyclic Petri nets is used.