Let ( P(n) ) be the minimum number of hashes needed to prove a leaf’s inclusion. Since each internal node covers disjoint subsets, a binary tree yields ( P(n) = \lceil \log_2 n \rceil ). A ( m )-ary tree would give ( \lceil \log_m n \rceil ) but at the cost of larger proofs per level (each sibling set size ( m-1 )), so total proof bits are ( (m-1) \cdot \lceil \log_m n \rceil \cdot k ). Minimizing over ( m ), the binary case (( m=2 )) minimizes total bits for proof transmission.
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