> For the complete documentation index, see [llms.txt](https://battlezips.gitbook.io/battlezips/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://battlezips.gitbook.io/battlezips/theory/primitives/merkle-trees/sparse-merkle-trees.md). # Sparse Merkle Trees ## Merkle Trees. Great for proving inclusion but useless for saying what isn't included In a previous section we described how a Merkle Tree proof of inclusion works. It is a fantastic tool for proving that a specific piece of data is part of a tree. Something that may have come to mind however is whether or not it is possible to prove that a piece of data is not contained within a Merkle Tree. With a standard tree this is not possible but it turns out that a Sparse Merkle Tree does precisely that! ## Sparse Merkle Tree A Sparse Merkle Tree is similar to a standard one but there is a key difference in the fact that the nodes are indexed. Proof of inclusion is possible in this implementation but so is proof of non-inclusion. To prove that a value is not contained within a tree we must prove that the leaf where the value would be contains the value null