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Conversational Recommender Systems (CRS) proactively engage users in interactive dialogues to elicit user preferences and provide personalized recommendations. Existing methods train Reinforcement Learning (RL)-based agent with greedy action selection or sampling strategy, and may suffer from suboptimal conversational planning. To address this, we present a novel Monte Carlo Tree Search (MCTS)-based CRS framework SAPIENT. SAPIENT consists of a conversational agent (S-agent) and a conversational planner (S-planner). S-planner builds a conversational search tree with MCTS based on the initial actions proposed by S-agent to find conversation plans. The best conversation plans from S-planner are used to guide the training of S-agent, creating a self-training loop where S-agent can iteratively improve its capability for conversational planning. Furthermore, we propose an efficient variant SAPIENT for trade-off between training efficiency and performance. Extensive experiments on four benchmark datasets validate the effectiveness of our approach, showing that SAPIENT outperforms the state-of-the-art baselines. Our code and data are accessible through https://github.com/ninglab/SAPIENT.