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Building and verification of Partial Merkle Tree. Partial Merkle Tree is a minimal tree needed to check that a given set of leaves belongs to a full Merkle Tree.

Example of Merkle tree with 5 leaves.

             h15
          /       \
         h14       h55
        /  \      /  \
      h12  h34   h50 h00
     / \   / \   / \  / \
    l1 l2 l3 l4 l5 0 0  0

l* denote hashes of leaves, h* - hashes of nodes below. 0 denotes zero hash, we use it to pad not full binary trees, so the number of leaves is always a power of 2.

Example of Partial tree based on the tree above.

           ___
        /       \
        _        _
      /  \      /  \
    h12   _     _   h00
         / \   / \
        I3 l4 I5 0

We want to check l3 and l5 - now turned into IncudedLeaf (I3 and I5 above). To verify that these two leaves belong to the tree with a hash root h15 we need to provide a Merkle branch (or partial tree). In our case we need hashes: h12, l4, 0 and h00. Verification is done by hashing the partial tree to obtain the root and checking it against the obtained h15 hash. Additionally we store included hashes used in calculation and compare them to leaves hashes we got (there can be a difference in obtained leaves ordering - that's why it's a set comparison not hashing leaves into a tree). If both equalities hold, we can assume that l3 and l5 belong to the transaction with root h15.