Escape the Tree

Chef is trying to make his birthday party adventurous. He recently heard about *escape the room* games, so he wants to prepare his own version of such a game. First, Chef will lock his friends in the root of a full binary tree with depth $h$, i.e. with $2^{h+1}-1$ nodes (numbered $1$ through $2^{h+1}-1$). The root has number $1$, but the other nodes may be numbered arbitrarily. His friends can