# Merkle mountain range
[](https://crates.io/crates/ckb-merkle-mountain-range)
A generalized merkle mountain range implementation.
## Construct
``` txt
# An 11 leaves MMR
14
/ \
6 13
/ \ / \
2 5 9 12 17
/ \ / \ / \ / \ / \
0 1 3 4 7 8 10 11 15 16 18
```
In MMR, we use the insertion order to reference leaves and nodes.
we inserting a new leaf to MMR by the following:
1. insert leaf or node to next position.
2. if the current position has a left sibling, we merge the left and right nodes to produce a new parent node, then go back to step 1 to insert the node.
For example, we insert a leaf to the example MMR:
1. insert leaf to next position: `19`.
2. now check the left sibling `18` and calculate parent node: `merge(mmr[18], mmr[19])`.
3. insert parent node to position `20`.
4. the node `20` also has a left sibling `17`, calculate parent node: `merge(mmr[17], mmr[20])`.
5. insert new node to next position `21`.
6. the node `20` have no left sibling, complete the insertion.
Example MMR after insertion of a new leaf:
``` txt
14
/ \
6 13 21
/ \ / \ / \
2 5 9 12 17 20
/ \ / \ / \ / \ / \ / \
0 1 3 4 7 8 10 11 15 16 18 19
```
## Merkle root
An MMR is constructed by one or more sub merkle trees (or mountains). Each sub merkle tree's root is a peak in MMR, we calculate the MMR root by bagging these peaks from right to left.
For example, we have a MMR with 3 peaks: `14, 17, 18`, we bagging thses peaks from right to left to get the root: `merge(merge(mmr[18], mmr[17]), mmr[14])`.
## Merkle proof
The merkle proof is an array of hashes constructed by the follows parts:
1. A merkle proof from the leaf's sibling to the peak that contains the leaf.
2. A hash that bagging all right-hand side peaks, skip this part if no right-hand peaks.
3. Hashes of all left-hand peaks, the from right to left, skip this part if no left-hand peaks.
We can reconstruct the merkle root from the proofs. Pre calculate the peaks positions from the size of MMR may help us do the bagging.
## References
* [Merkle mountain range](https://github.com/opentimestamps/opentimestamps-server/blob/master/doc/merkle-mountain-range.md)
* [Grin Doc](https://github.com/mimblewimble/grin/blob/master/doc/mmr.md#structure)