Building a Merkle tree requires ( n-1 ) hash operations for internal nodes, i.e., ( O(n) ) time. Verifying a single leaf proof requires ( \lceil \log_2 n \rceil ) hashes, i.e., ( O(\log n) ) time. Thus, Merkle trees offer an exponential improvement in verification time compared to linear lists (( O(n) ) verification) and match the lower bound for membership proofs in a static, authenticated data structure.
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For dynamic updates (changing one leaf), recompute path from leaf to root: Building a Merkle tree requires ( n-1 )