## Table of contents

## What is a Binary Tree?

A binary tree is a tree-like data structure in which each node has at most two children, referred to as left child and right child. The first node in the tree is called the root node. If a node has no children, it is referred to as a leaf node. Each node in a binary tree can have zero, one, or two children.

Binary trees are used to store data in a hierarchical manner, allowing for efficient searching and sorting. They are used in various applications, such as search trees, expression trees, and Huffman trees.

## Types of Binary Trees

There are several types of binary trees, including full binary trees, complete binary trees, and balanced binary trees.

### Full Binary Trees

A full binary tree is a binary tree in which every node has either zero or two children. In other words, every node in a full binary tree has either two children or no children at all.

### Complete Binary Trees

A complete binary tree is a binary tree in which every level, except possibly the last level, is completely filled, and all nodes are as far left as possible. In other words, a complete binary tree is a binary tree in which all nodes have either two children or no children, except for the leaf nodes on the last level, which are filled from left to right.

### Balanced Binary Trees

A balanced binary tree is a binary tree in which the height of the left and right subtrees of every node differ by at most one. In other words, a balanced binary tree is a binary tree in which the difference in height between the left and right subtrees of any node is at most one.

## Properties of Binary Trees

Binary trees have several properties, including height, depth, degree, fullness, and completeness.

### Height

The height of a binary tree is the number of edges on the longest path from the root node to a leaf node. In other words, the height of a binary tree is the length of the longest path from the root node to a leaf node.

### Depth

The depth of a node in a binary tree is the number of edges from the root node to that node. In other words, the depth of a node in a binary tree is the length of the path from the root node to that node.

### Degree

The degree of a node in a binary tree is the number of children that the node has. In other words, the degree of a node in a binary tree is the number of nodes that are directly connected to that node.

### Fullness

A binary tree is said to be full if every node has either zero or two children.

### Completeness

A binary tree is said to be complete if all levels, except possibly the last level, are completely filled, and all nodes are as far left as possible.

## Operations on Binary Trees

Binary trees support various operations, such as insertion, deletion, and traversal.

### Insertion

Insertion is the process of adding a new node to the binary tree. To insert a new node, we start at the root node and compare the value of the new node with the value of the current node. If the new node’s value is less than the current node’s value, we go to the left subtree. If the new node’s value is greater than the current node’s value, we go to the right subtree. We continue this process until we reach a node that has no children, and then we add the new node as its child.

### Deletion

Deletion is the process of removing a node from the binary tree. There are several cases to consider when deleting a node from a binary tree, depending on whether the node is a leaf node, has one child, or has two children.

### Traversal

Traversal is the process of visiting each node in the binary tree once. There are three main methods for traversing a binary tree:

- In-order traversal: visits the left subtree, then the root node, then the right subtree.
- Pre-order traversal: visits the root node, then the left subtree, then the right subtree.
- Post-order traversal: visits the left subtree, then the right subtree, then the root node.

## Applications of Binary Trees

Binary trees have many applications, including search trees, expression trees, and Huffman trees.

### Search Trees

Search trees are binary trees that are used to store data in a way that allows for efficient searching. In a search tree, the left subtree contains values that are less than the root node’s value, and the right subtree contains values that are greater than the root node’s value. This allows for efficient searching by narrowing down the search space with each comparison.

### Expression Trees

Expression trees are binary trees that are used to represent arithmetic expressions. In an expression tree, each node represents an operator, and its children represent the operands. This allows for efficient evaluation of arithmetic expressions.

### Huffman Trees

Huffman trees are binary trees that are used for data compression. In a Huffman tree, each leaf node represents a symbol, and its weight represents the frequency of that symbol in the data. The tree is constructed by repeatedly combining the two nodes with the lowest weights until all nodes are combined into a single node.

## Advantages and Disadvantages of Binary Trees

Binary trees have several advantages, such as efficient searching and sorting, and are used in various applications, such as search trees, expression trees, and Huffman trees. However, binary trees also have some disadvantages, such as their limited height and their potential for becoming unbalanced.

## Conclusion

Binary trees are a fundamental data structure used in computer science for storing data in a hierarchical manner. They come in various types, such as full binary trees, complete binary trees, and balanced binary trees. Binary trees have many applications, including search trees, expression trees, and Huffman trees. While binary trees have several advantages, they also have some disadvantages, such as their limited height and their potential for becoming unbalanced.

## Related Topics

- Tree
- Binary Tree
- Heaps
- Hashing
- Advanced Data Structures
- Database Systems
- Operating Systems
- Computer Networks