Binary search trees are generally familiar to those who are interested in data structures and algorithms. These structures are widely used in computer science because they allow efficient searching, insertion, and deletion of items. We’ll go in-depth on binary search trees in this essay, covering both fundamental and cutting-edge ideas.

**Table of Contents**

- Introduction
- Properties of Binary Search Trees
- BST Rules
- Height and Depth of a BST
- Balanced BSTs

- Operations on Binary Search Trees
- Searching
- Insertion
- Deletion
- Traversal

- Variations of Binary Search Trees
- AVL Trees
- Red-Black Trees
- Splay Trees

- Use Cases for Binary Search Trees
- Database Indexing
- Searching Algorithms
- Sorting Algorithms

- Conclusion

**1. Introduction**

A hierarchical data structure for storing and organizing data is a binary search tree. Each node in the structure, known as the left child and the right child, has a maximum of two children. Additionally, each node has a key value that may be used to compare it to other nodes in the tree. The key value of the left child is always less than that of its parent, but the key value of the right child is continuously greater than or equal to that of its parent.

Binary search trees are commonly used in computer science due to their efficient searching and sorting capabilities. They have several beneficial qualities, which we shall examine in the following part.

**2. Properties of Binary Search Trees**

**BST Rules**

As previously mentioned, each node in a binary search tree has at most two children. Additionally, in order to preserve the integrity of the tree, two important guidelines must be followed:

- The key value of the left child is always less than the key value of its parent.
- The key value of the right child is always greater than or equal to the key value of its parent.

**Height and Depth of a BST**

The maximum number of edges from the root node to a leaf node determines the height of a binary search tree. The quantity of edges connecting a node to its root determines that node’s depth. A binary search tree with just the root (one node) has a height of 0. A binary search tree’s height may be determined using the formula below:

height = max(left_height, right_height) + 1

where the heights of the left and right subtrees, respectively, are left_height and right_height.

**Balanced BSTs**

A binary search tree is considered balanced if the height of its left and right subtrees differ by at most one. A balanced binary search tree ensures that operations such as searching, insertion, and deletion take O(log n) time, where n is the number of nodes in the tree.

**3. Operations on Binary Search Trees**

**Searching**

Searching for a node in a binary search tree involves comparing the value of the node with the key value being searched for. If the value is equal to the key, the node has been found. The search proceeds in the appropriate subtree if the value is smaller than the key. If the value is greater than the key,

the search continues in the left subtree. The search algorithm will continue until the key value is found or the search reaches a leaf node, indicating that the key value is not present in the tree.

**Insertion**

Inserting a new node into a binary search tree involves finding the proper position for the new node in the tree based on its key value. The algorithm starts at the root node and compares the key value of the new node with the key value of the current node. If the new key value is less than the current node’s key value, the algorithm continues down the left subtree. If the new key value is greater than the current node’s key value, the algorithm continues down the right subtree. Up until the algorithm reaches a leaf node, this step is repeated. The new node is now put into the tree as a child of the leaf node.

**Deletion**

Deleting a node from a binary search tree involves finding the node to be deleted and then removing it from the tree. If the node to be deleted is a leaf node, it is simply removed from the tree.A child is promoted to take the position of the removed node in the tree if the node to be deleted has just one child. If the node to be deleted has two children, the algorithm finds the node’s successor, which is the node with the smallest key value in the node’s right subtree. The successor node is then swapped with the node to be deleted, and the original node is removed from the tree.

**Traversal**

Traversal is the process of visiting every node in a binary search tree in a specific order. There are three common traversal methods:

- In-order traversal: visits the current node, the left subtree, and then the right subtree.
- Pre-order traversal: visits the current node before moving on to the left and right subtrees.
- Post-order traversal: visits the current node, then the left subtree, and finally the right subtree.

**4. Variations of Binary Search Trees**

**AVL Trees**

AVL trees are a type of self-balancing binary search tree in which the heights of the left and right subtrees differ by at most one. When a node is inserted into or deleted from an AVL tree, the tree is rebalanced to maintain this property. The Adelson-Velsky and Landis (AVL) trees are so-called in honor of their creators.

**Red-Black Trees**

Red-black trees are another type of self-balancing binary search tree. They guarantee that the tree’s height will never exceed 2log2(n+1), where n is the number of nodes in the tree. Because each node in red-black trees is either red or black, they get their name.

**Splay Tree**

It is a type of self-adjusting binary search tree that reorganizes itself after each access operation to optimize future access times. Splay trees are named after their main operation, “splaying”, which moves a node to the root of the tree.

**5. Use Cases for Binary Search Trees**

**Database Indexing**

Binary search trees are commonly used in databases to create indexes for efficient searching and sorting. A database index is a type of data structure that accelerates operations on database tables for data retrieval.

**Searching Algorithms**

In search algorithms like binary search, which is used to locate an item in a sorted array or list, binary search trees are frequently utilized. The A* method, used in pathfinding, may alternatively be implemented using binary search trees.

**Sorting Algorithms**

Binary search trees can be used in sorting algorithms, such as the heap sort algorithm. The heap sort algorithm uses a binary heap

to sort an array of elements.

**Symbol Tables**

Symbol tables are data structures that associate a value with a key. Binary search trees can be used to implement symbol tables, with each node containing a key-value pair.

**Dynamic Programming**

By dissecting optimization issues into smaller subproblems, dynamic programming is an approach for finding solutions to optimization problems. Dynamic programming algorithms like the optimum binary search tree algorithm may be implemented using binary search trees.

**Game Trees**

In game theory, game trees are used to illustrate various game outcomes. Game trees can be implemented using binary search trees, where each node represents a potential game state.

**6. Conclusion**

In conclusion, binary search trees are a fundamental data structure in computer science. They offer efficient searching, insertion, and deletion operations, and can be used in a variety of applications, including database indexing, searching and sorting algorithms, symbol tables, dynamic programming, and game trees. Different variations of binary search trees, such as AVL trees, red-black trees, and splay trees, offer additional properties and optimizations. Understanding binary search trees and their applications is an essential part of any computer science education.

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