Large Language Models (LLMs) have shown remarkable reasoning capabilities through Reinforcement Learning with Verifiable Rewards (RLVR) methods. However, a key limitation of existing approaches is that rewards defined at the full trajectory level provide insufficient guidance for optimizing the intermediate steps of a reasoning process. To address this, we introduce, a novel method that estimates the mathematical expectations of rewards at various reasoning steps using tree sampling. Unlike prior methods that rely on a separate step reward model, directly estimates these rewards through this sampling process. Building on the group-relative reward training mechanism of GRPO, innovatively computes rewards based on step-level groups generated during tree sampling. This advancement allows to produce fine-grained and dense reward signals, significantly enhancing the learning process and overall performance of LLMs. Experimental results demonstrate that our algorithm substantially improves the average Pass@1 accuracy of Qwen-2. 5-Math on test benchmarks, increasing it from 19. 0\% to 35. 5\%. Furthermore, significantly outperforms GRPO by 2. 9\% in performance while simultaneously reducing the average response length by 18. 1\%, showcasing its effectiveness and efficiency. Our code will be available at https: //github. com/yangzhch6/TreeRPOhttps: //github. com/yangzhch6/TreeRPO.
Yang et al. (2025) studied this question.