🌳 Binary Trees – Comprehensive Study Guide

A **Binary Tree** is a **hierarchical data structure** where each node has **at most two children**. It is widely used in **searching, graph traversal, parsing expressions, and AI algorithms**.

📌 Why are Binary Trees Important?

  • 🚀 **Efficient Searching** – Used in **Binary Search Trees (BSTs)**.
  • ⚡ **Optimized Storage** – Used in **heap-based priority queues**.
  • 🔄 **Traversal in Various Orders** – Used in **expression parsing, Huffman coding, etc.**
  • 📂 **Used in File Systems & Databases** – Hierarchical data management.

📚 Binary Tree Terminologies

  • 🌳 **Root** – The topmost node of a tree.
  • 🍃 **Leaf Node** – A node with no children.
  • 🔄 **Parent & Child** – A node **above** is called a **parent**, and the node **below** is called a **child**.
  • 🔢 **Degree** – The number of children a node has.
  • 📏 **Height of a Node** – The number of edges on the longest path from a node to a leaf.

🛠️ Representation of a Binary Tree

1️⃣ **Using Arrays (For Complete Binary Trees)**

A complete binary tree can be represented as an **array**, where:

  • **Left child** of `i` is at `2*i + 1`.
  • **Right child** of `i` is at `2*i + 2`.

                int tree[] = {1, 2, 3, 4, 5, 6, 7}; // Binary tree stored in an array

2️⃣ **Using Pointers (Dynamic Representation)**


                struct Node {
                    int data;
                    Node* left;
                    Node* right;
                };

📌 Types of Binary Trees

  • ✅ **Full Binary Tree** – Every node has **0 or 2 children**.
  • ✅ **Complete Binary Tree** – All levels are completely filled except possibly the last.
  • ✅ **Perfect Binary Tree** – All internal nodes have **two children**, and all leaves are at the same level.
  • ✅ **Balanced Binary Tree** – The height of the tree is **logarithmic** in terms of the number of nodes.

🔄 Tree Traversal Techniques

📌 **1. Depth-First Traversal (DFS)**

  • ➡️ **Preorder (Root → Left → Right)**
  • ➡️ **Inorder (Left → Root → Right) – Used in BSTs**
  • ➡️ **Postorder (Left → Right → Root) – Used in expression trees**

📌 **2. Breadth-First Traversal (BFS)**

  • 🔢 **Level Order Traversal (Node by node level-wise)**

🔍 Binary Search Tree (BST)

A **Binary Search Tree (BST)** is a special type of binary tree where:

  • ✔️ Left subtree **contains smaller values**.
  • ✔️ Right subtree **contains larger values**.
  • ✔️ Searching and insertion take **O(log n)** time.

⚖️ Balanced Binary Trees

  • 📌 **AVL Trees** – Self-balancing BSTs.
  • 📌 **Red-Black Trees** – Used in **self-balancing operations**.
  • 📌 **B-Trees** – Used in **database indexing**.

⏳ Time Complexity Analysis

Operation Best Case Worst Case
Search O(log n) O(n)
Insertion O(log n) O(n)
Deletion O(log n) O(n)

🌍 Real-World Applications of Binary Trees

  • 📂 **File System Management** – Organizing files in directories.
  • 🔍 **Binary Search Trees (BST)** – Efficient **searching & sorting**.
  • 🛒 **E-commerce Platforms** – Recommendation systems.
  • 🤖 **Artificial Intelligence (AI)** – Decision Trees.

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