Cyclic behavior of maxima in a hierarchical summation scheme

Let i.i.d. symmetric Bernoulli random variables be associated to the edges of a binary tree having $n$ levels. To any leaf of the tree, we associate the sum of variables along the path connecting the leaf with the tree root. Let $M_n$ denote the maximum of all such sums. We prove that, as $n$ grows, the distributions of $M_n$ approach some helix in the space of distributions. Each element of this helix is an accumulation point for the shifts of distributions of $M_n$.