Each edge of the standard rooted binary tree is equipped with a random weight; weights
are independent and identically distibuted. The value of a vertex is the sum of the weights
on the path from the root to the vertex. We wish to search the tree to find a vertex of
large weight. A very natural conjecture of Aldous states that, in the sense of stochastic
domination, an obvious greedy algorithm is best possible. We show that this conjecture is
false. We prove, however, that in a weaker sense there is no significantly better algorithm.