To prove that a block ( d_i ) is part of ( D ) given only ( R ), one provides ( O(\log n) ) sibling hashes. Let ( \pi_i ) be the set of siblings along the path from leaf ( L_i ) to root. Verification recomputes the root and checks equality.
One of the most beautiful mathematical properties of a Merkle tree is rarely discussed outside of formal proofs: . Matematicka Analiza Merkle 19.pdf
If you solve that for typical hardware (say, SHA-256 at 1µs, network at 100µs per hash), the optimal $b$ hovers around 16–22. The number 19 is the mathematical sweet spot for a specific era of computing (late 2010s, early 2020s). To prove that a block ( d_i )