# What is a binary search tree, and how does it differ from a regular binary tree?

### **Binary Search Tree (BST):**

A **Binary Search Tree (BST)** is a type of binary tree where nodes are organized based on their key values to maintain a specific order. It adheres to the following properties:

1. **Left Subtree Rule**: For any node, all values in its left subtree are less than the value of the node.
2. **Right Subtree Rule**: For any node, all values in its right subtree are greater than the value of the node.
3. **Uniqueness**: Typically, BSTs do not allow duplicate values (though this can vary depending on implementation).

These properties ensure that a **BST is sorted**, which allows for efficient searching, insertion, and deletion operations.

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### **Binary Tree (General Definition):**

A **Binary Tree** is a tree data structure where each node has at most two children. It does not impose any ordering constraints on the values of the nodes.

- **Structure**: Each node can have a **left child**, a **right child**, or both (or no children).
- **No Value Order**: The values in a binary tree are not arranged in any specific order.

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### **Key Differences Between Binary Tree and Binary Search Tree:**

|**Aspect**|**Binary Tree**|**Binary Search Tree**|
|---|---|---|
|**Node Value Order**|No specific order for node values.|Nodes are organized by the BST property (left < root < right).|
|**Searching Efficiency**|Searching is linear in worst case (O(n)).|Searching is efficient (O(log n)) if balanced.|
|**Insertion Rule**|Nodes can be added in any order.|Nodes are inserted based on their value to maintain the BST property.|
|**Duplicates**|Allows duplicates freely.|May or may not allow duplicates, depending on implementation.|
|**Applications**|Used for general tree structures.|Used for sorted collections or when efficient searching is required.|

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### **Example:**

#### Binary Tree:

A binary tree with no ordering constraints:
```
		10
       /  \
      20   30
     / 
    40
```
- Node values are not in any specific order.

#### Binary Search Tree:

A binary search tree:
```
		10
       /  \
      5    20
     / \   / \
    2   7 15  25
```

- For every node:
    - Left subtree values < Node value.
    - Right subtree values > Node value.

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### **Advantages of BST Over a General Binary Tree:**

1. **Efficient Searching**: Searching in a BST takes $O(h)$, where $h$ is the height of the tree. In a balanced BST, $h=O(log⁡n)$.
2. **Ordered Traversal**: In-order traversal of a BST produces the nodes in sorted order.
3. **Efficient Range Queries**: BSTs can efficiently find all elements within a specific range.

### **Disadvantages of BST:**

1. If unbalanced (e.g., when nodes are inserted in sorted order), the BST can degrade to a linked list, resulting in $O(n)$ time for operations.
2. Maintaining balance (e.g., in AVL or Red-Black Trees) adds complexity.