Search cost for a nearly optimal path in a binary tree

Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p ≤ 1/2. How many of these Bernoullis one must look at in order to find a path of length n from the root which maximizes, up to a factor of 1− , the sum of the Bernoullis along the path? In the case, p = 1/2 (the critical value for nontriviality), it is shown to take Θ( −1n) steps. In the case p < 1/2, the number of steps is shown to be at least n · exp(const −1/2). This last result matches the known upper bound from [Ald98] in a certain family of subcases.