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rln-research/merkle-tree-update.md
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# Solution Overview
In a Merkle Tree with the capacity of `n` leaves , one can compute the root of the tree by maintaining the root nodes of log(n) number of complete Merkle trees.
> In a Merkle Tree with the capacity of `n` leaves , one can compute the root of the tree by maintaining the root nodes of log(n) number of complete Merkle trees.
We use this fact and define `F` to be an array of size log(n) holding the root of complete Merkle trees (all positioned on the left side of the tree) for levels `[0, d=log(n)+1]`. Each entry of `F` at position `i` holds a pair `(index, H)` where H is the root of the complete subtree at level `i`, and `index` indicates the index of the leaf node whose insertion resulted in `H`.
We use the preciding observation and define `F = [(L0,H0,index0), ..., (Ld,Hd,indexd)]` to be an array of size log(n)+1 holding the root of the complete Merkle trees (all positioned on the left side of the tree) for levels `[0, d=log(n)+1]`. Each entry of `F` holds a tuple `(L,index, H)` `H`is the root of the complete subtree at level `L`, and `index` indicates the index of the leaf node whose insertion resulted in `H`.
`F = [(L0,H0,index0), ..., (Ld,Hd,indexd)]`
For the Merkle Tree shown in Figure below, `F = [(L0, N12, leaf5), (L1, N6, leaf6), (L2, N2, leaf4), (L3, N1, leaf6)]` is highlighted in green. Note that none of the gray nodes are stored as part of `F`.
![Tree](F.png =250x250)
![Tree](F.png)
## Deletion
## Computing the root after deletion
Consider the deletion of the `leafi` with the authentication path / (membership proof) of the follwoing form `authpath = [(L0,H0), ..., (Ld,Hd)]`.
The authentication path of `leaf2` is illustrated in the following figure. `authpath2 = [(L0, N8), (L1,N5), (L2,N3), (L3,N1)]`.
Consider the deletion of the `leafi` with the authentication path / (membership proof) of the follwoing form `authpath = [(L0,H0), ..., (Ld,Hd)]` where `H` values represent the value of the merkle tree nodes at the indicated level `L`.
The authentication path of `leaf2` is illustrated in the following figure and consists of `authpath2 = [(L0, N8), (L1,N5), (L2,N3), (L3,N1)]`.
![authPath](authPath.png)