# Patricia Merkle Tree

A Patricia Merkle Tree, also known as a Modified Merkle Patricia Tree, is a data structure used in blockchain technology to efficiently store and retrieve large amounts of data in a secure and tamper-proof manner. It is a type of Merkle tree that is designed specifically for use in blockchain systems.

In a Patricia Merkle Tree, each node represents a key-value pair, where the key is a hash of the data being stored, and the value is the actual data. The nodes are organized in a way that makes it easy to search and retrieve data. Unlike a regular Merkle tree, the nodes in a Patricia Merkle Tree are not limited to two children per node.

The name "Patricia" stands for "Practical Algorithm to Retrieve Information Coded in Alphanumeric", which refers to the efficient data retrieval capabilities of the tree. By using a Patricia Merkle Tree, blockchain systems can store and retrieve data more efficiently, which helps to improve the performance of the system.

### Example of a Patricia Merkle Tree in the Context of a Blockchain

Suppose we have a blockchain that stores transaction data. Each transaction has a unique ID, which we can use as the key for our Patricia Merkle Tree. Each transaction also has a set of inputs and outputs, which we can use as the value for the corresponding key.

To construct the Patricia Merkle Tree, we first hash each transaction ID using a cryptographic hash function, such as SHA-256. We then insert each hash-value pair into the tree. The tree is organized in a way that allows us to quickly search and retrieve transactions based on their ID.

Suppose we want to look up a specific transaction with ID "abc123". We would start at the root of the tree and follow the path corresponding to the binary representation of the hash of "abc123". Along the way, we would encounter various nodes in the tree, each of which represents a prefix of the hash. We would continue down the path until we reach a leaf node, which contains the actual transaction data.

The Patricia Merkle Tree allows us to efficiently store and retrieve large amounts of transaction data, while also ensuring that the data is secure and tamper-proof. If any of the transaction data is changed, the hash values would also change, and the tree would be invalidated, making it easy to detect any attempts at tampering with the data.

### Example

In this example, the tree has a root node with two children, labeled "01" and "04". The "01" node has two children, labeled "012" and "013", and the "04" node has one child, labeled "046". The "012" node has two children, labeled "0123" and "0125", and the "046" node has two children, labeled "0468" and "0469".

Each node in the tree represents a key-value pair, where the key is a hash of the data being stored, and the value is the actual data. In this case, the keys are the numbers in square brackets, and the values are not shown.

To look up a specific value in the tree, we would start at the root and follow the path corresponding to the binary representation of the hash of the key we are looking for. For example, to look up the value corresponding to key "0123", we would start at the root, follow the path labeled "01", then "012", and finally "0123", until we reach the leaf node containing the corresponding value.

### Example 2

Let's say we have the following key-value pairs that we want to store in a Patricia Merkle Tree:

Here's how the Patricia Merkle Tree would look like:

In this example, the root node of the tree is the hash of the entire tree. Each node in the tree is a hash of its children nodes and its own value. The edges in the tree represent the prefix of the key that is shared by the child node and its parent node.

Note that in a Patricia Merkle Tree, the keys are not stored in the tree explicitly, but rather the edges of the tree represent the keys.

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