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This paper proposes a control method for the multi-agent pickup and delivery problem (MAPD problem) by extending the priority inheritance with backtracking (PIBT) method to make it applicable to more general environments. PIBT is an effective algorithm that introduces a priority to each agent, and at each timestep, the agents, in descending order of priority, decide their next neighboring locations in the next timestep through communications only with the local agents. Unfortunately, PIBT is only applicable to environments that are modeled as a bi-connected area, and if it contains dead-ends, such as tree-shaped paths, PIBT may cause deadlocks. However, in the real-world environment, there are many dead-end paths to locations such as the shelves where materials are stored as well as loading/unloading locations to transportation trucks. Our proposed method enables MAPD tasks to be performed in environments with some tree-shaped paths without deadlock while preserving the PIBT feature; it does this by allowing the agents to have temporary priorities and restricting agents' movements in the trees. First, we demonstrate that agents can always reach their delivery without deadlock. Our experiments indicate that the proposed method is very efficient, even in environments where PIBT is not applicable, by comparing them with those obtained using the well-known token passing method as a baseline.