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same way producing just 32 bytes of data as the single merkle root. In the diagram below, you will see a tree built from 16 transactions. Note that while the root looks bigger than the leaf nodes in the diagram, it is the exact same size, just 32 bytes. Whether there is one transaction or a hundred thousand transactions in the block, the merkle root always summarizes them into 32 bytes: Figure 7-4. A Merkle Tree summarizing many data elements To prove that a specific transaction is included in a block, a node only needs to produce log2(N) 32-byte hashes, constituting an authentication path or merkle path connecting the specific transaction to the root of the tree. This is especially important as the number of transactions increases, because the base-2 logarithm of the number of transactions increases much more slowly. This allows bitcoin nodes to efficiently produce paths of ten or twelve hashes (320-384 bytes) which can provide proof of a single transaction out of more than a thousand transactions in a megabyte sized block. In the example below, a node can prove that a transaction K is included in the block by producing a merkle path that is only four 32-byte hashes long (128 bytes total). The path consists of the four hashes (noted in blue in the diagram below) HL, HIJ, HMNOP and HABCDEFGH. With those four hashes provided as an authentication path, any node can prove that HK (noted in green in the diagram below) is included in the merkle root by computing four addi‐ tional pair-wise hashes HKL, HIJKL and HIJKLMNOP (outlined in a dotted line in the diagram below) that lead to the merkle root. 172 | Chapter 7: The BlockchainPDF Image | Mastering Bitcoin
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