# Merkle¶

For an overview of Merkle trees, see wikipedia.

There are two types of Merkle trees used in Tendermint.

**IAVL+ Tree**: An immutable self-balancing binary tree for persistent application state**Simple Tree**: A simple compact binary tree for a static list of items

## IAVL+ Tree¶

The purpose of this data structure is to provide persistent storage for key-value pairs (e.g. account state, name-registrar data, and per-contract data) such that a deterministic merkle root hash can be computed. The tree is balanced using a variant of the AVL algorithm so all operations are O(log(n)).

Nodes of this tree are immutable and indexed by its hash. Thus any node serves as an immutable snapshot which lets us stage uncommitted transactions from the mempool cheaply, and we can instantly roll back to the last committed state to process transactions of a newly committed block (which may not be the same set of transactions as those from the mempool).

In an AVL tree, the heights of the two child subtrees of any node differ by at most one. Whenever this condition is violated upon an update, the tree is rebalanced by creating O(log(n)) new nodes that point to unmodified nodes of the old tree. In the original AVL algorithm, inner nodes can also hold key-value pairs. The AVL+ algorithm (note the plus) modifies the AVL algorithm to keep all values on leaf nodes, while only using branch-nodes to store keys. This simplifies the algorithm while minimizing the size of merkle proofs

In Ethereum, the analog is the Patricia trie. There are tradeoffs. Keys do not need to be hashed prior to insertion in IAVL+ trees, so this provides faster iteration in the key space which may benefit some applications. The logic is simpler to implement, requiring only two types of nodes – inner nodes and leaf nodes. The IAVL+ tree is a binary tree, so merkle proofs are much shorter than the base 16 Patricia trie. On the other hand, while IAVL+ trees provide a deterministic merkle root hash, it depends on the order of updates. In practice this shouldn’t be a problem, since you can efficiently encode the tree structure when serializing the tree contents.

## Simple Tree¶

For merkelizing smaller static lists, use the Simple Tree. The transactions and validation signatures of a block are hashed using this simple merkle tree logic.

If the number of items is not a power of two, the tree will not be full and some leaf nodes will be at different levels. Simple Tree tries to keep both sides of the tree the same size, but the left side may be one greater.

```
Simple Tree with 6 items Simple Tree with 7 items
* *
/ \ / \
/ \ / \
/ \ / \
/ \ / \
* * * *
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
* h2 * h5 * * * h6
/ \ / \ / \ / \ / \
h0 h1 h3 h4 h0 h1 h2 h3 h4 h5
```

### Simple Tree with Dictionaries¶

The Simple Tree is used to merkelize a list of items, so to merkelize a
(short) dictionary of key-value pairs, encode the dictionary as an
ordered list of `KVPair`

structs. The block hash is such a hash
derived from all the fields of the block `Header`

. The state hash is
similarly derived.