KöMaL 2013 A592

Color the vertices of a binary tree of depth $2^{2n}$ as follows. In the initial position let all vertices be white. Take the vertices one by one in a random order. Color the current vertex to red, except if from that vertex there starts a path of length $n$ downwards (away from the root), whose other vertices are already red. Let $p$($n$) be the probability that during the procedure, the root remains white. Determine $\lim_{n\to\infty} p(n)$.