Every decision tree has an influential variable

We prove that for any decision tree calculating a boolean function f : {-1,1}(n) -> {-1,1}, Var[f] <= Sigma(n)(i=1)delta(i) Inf(i)(f), where delta(i) is the probability that the ith input variable is read and Inf(i)(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 {-1,1}(n). It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the 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 (1)/(d) The only previous nontrivial lower bound known was Omega(d2(-d)). Our inequality 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 application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least Omega(v(4/3)/p(1/3)), where v is the number of vertices and p <= (1)/(2) is the critical threshold probability. This supersedes the milestone Omega(v(4/3)) bound of Hajnal [13] and is sometimes superior to the best known lower bounds of Chakrabarti-Khot [9] and Friedgut-Kahn- Wigderson [11].