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Edit: I prefer to formulate first the problem as Fedor Petrov suggests in the comments: We are given a multiset $F$, initially containing only the single integer $h$. Sequentially, at each time step, ...

Let $B_h$ be the perfect binary tree having height $h$ (i.e. the binary tree with height $h$ in which all interior nodes have two children and all leaves have the same depth or same level). For any ...