Tree-search decoding is an effective form of test-time scaling for large language models (LLMs), but real-world deployment often imposes a fixed per-query token budget that varies across settings. Existing tree-search policies are largely budget-agnostic, treating the budget merely as a termination condition, thereby risking late-stage over-branching or premature termination. We propose Budget-Guided MCTS (BG-MCTS), a tree-search decoding algorithm that aligns its search policy with the remaining token budget: it starts with broad exploration, then prioritizes refinement and answer completion as the remaining budget decreases while reducing late-stage branching from shallow nodes. BG-MCTS consistently outperforms budget-agnostic tree-search baselines across inference budgets on mathematical reasoning benchmarks and an additional physics reasoning benchmark with open-weight LLMs.