(data structure)

**Definition:**
A *tree* (usually a *binary tree*) in which each *internal node* has a *hash* of all the information in the *leaf* nodes under it. Specifically, each internal node has a hash of the information in its *children*. Each leaf has a hash of the block of information it represents. All leaf nodes are at the same *depth*. All nodes are as far left as possible.

To add a block (leaf) to a *full* tree, a new *root* is created with the old root as its left child. Its right child is a degenerate tree (only left subtrees) all the way to the leaf level.

**Also known as** hash tree.

**Generalization** (I am a kind of ...)

*tree*.

**Aggregate child** (... is a part of or used in me.)

*hash function*.

**See also**
*Merkle tree [Wikipedia]*.

*Note:
Usually the branching is two, making it a binary tree, but it can be a k-ary tree. *

Merkle's paper describes a conceptually infinite (binary) tree with blocks (documents) at each internal node. To add a new block, use the next node in a *level-order traversal*. (A level-order traversal is the same order as a *breadth-first search* from the root.)

Given two copies of possibly-the-same information represented by Merkle trees, one can check whether the copies are the same by just comparing the hashes at the root. A single difference between the copies can be found by just comparing the hashes at internal nodes on a path to the different leaf. This is logarithmic in the number of leaves.

* A complete tree has all nodes as far left as possible, too, but every level is filled.*

Author: PEB

Average space required and complexity for search, traversal, insert, delete, and synchronize.

**Ralph C. Merkle**, *Method of providing digital signatures*, U.S. Patent 4309569, filed 5 September 1979. PDF image available at http://pdfpiw.uspto.gov/.piw?Docid=04309569. Text version available at http://patft.uspto.gov/netahtml/PTO/srchnum.htm enter 4309569 and click Search.

**Ralph C. Merkle**, *A Digital Signature Based on a Conventional Encryption Function*, in C. Pomerance (ed) Advances in Cryptology — CRYPTO ’87. Lecture Notes in Computer Science 293:369-378, 1988. doi:10.1007/3-540-48184-2_32

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 23 February 2018.

HTML page formatted Tue Feb 12 10:57:43 2019.

Cite this as:

Paul E. Black, "Merkle tree", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 23 February 2018. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/MerkleTree.html