diff --git a/README.md b/README.md
index 8fd0ab2..cfbc9f2 100644
--- a/README.md
+++ b/README.md
@@ -2,7 +2,64 @@
 
 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
 
-* https://github.com/mimblewimble/grin/blob/master/doc/mmr.md#structure
-* https://github.com/opentimestamps/opentimestamps-server/blob/master/doc/merkle-mountain-range.md
+* [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)