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Abstract We prove that for any decision tree calculating a booleanfunction f: f\Gamma 1; 1gn! f\Gamma 1; 1g, Var[f] ^ nX i=1 ffii Infi(f); where ffii is the probability that the ith input variable isread and Infi(f) is the influence of the ith variable on f.The variance, influence and probability are taken with respect to an arbitrary product measure on f\Gamma 1; 1gn. It fol-lows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal ofthe largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth d has a variable with influence at least 1d. The only pre-vious nontrivial lower bound known was \Omega (d2\Gamma d). Ourinequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees,decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an ap-plication of our results we give a very easy proof tha

Year: 2006
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