Tree search enhances multi-step reasoning in large language models (LLMs), yet prevailing methods expand every step with a fixed budget, irrespective of the marginal utility of expansion. We first show, across existing search algorithms, that continuations sampled from a shared intermediate state are largely duplicates and collapse to a few distinct answers. Critically, we find that the semantic dispersion of sibling actions predicts the redundancy of their downstream continuations, permitting a subtree's budget to be fixed before expansion. An information-theoretic analysis motivates this: treating the per-step budget as the Monte Carlo sample size for estimating a step's residual answer entropy, we derive a sufficient budget that shrinks with sibling dispersion. We accordingly propose $\textbf{Uncertainty-Aware Allocation}$ (UAA), a training-free framework that decouples value from uncertainty: values determine which candidates are promising, while uncertainty, estimated from sibling dispersion at no additional model cost, determines how many merit expansion, via uncertainty-aware pruning and uncertainty-aware budgeting. Across mathematical reasoning, logical reasoning, and planning benchmarks, UAA reduces token usage by up to $90\%$ and latency by up to $9\times$ at comparable accuracy, generalizing across four tree search algorithms and both open- and closed-source LLMs.

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