A binary tree is a type of data structure in which each node has at most two children, referred to as the left child and the right child. This hierarchical structure is used to store data in a way that makes it efficient to search, insert, and delete elements.

Key Characteristics of a Binary Tree

• Node: Each element in a binary tree is called a node. A node contains a value and may have a left child, a right child, or both.
• Root: The top node of a binary tree is called the root. It is the starting point of the tree.
• Children: The nodes connected to a node are called its children. In a binary tree, each node has at most two children.
• Leaf: A node with no children is called a leaf.
• Subtree: A subtree is any node and all its descendants in the binary tree.

Types of Binary Trees

• Full Binary Tree: A binary tree in which every node has 0 or 2 children.
• Perfect Binary Tree: A binary tree in which all interior nodes have two children and all leaves have the same depth.
• Complete Binary Tree: A binary tree in which all levels are completely filled except possibly for the last level, which is filled from left to right.
• Balanced Binary Tree: A binary tree in which the height of the left and right subtrees of any node differ by at most one.
• Binary Search Tree (BST): A binary tree in which the left child of a node contains a value less than the node's value, and the right child contains a value greater than the node's value.

Operations on a Binary Tree

• Insertion: Adding a new node to the tree at the appropriate position.
• Deletion: Removing a node from the tree and rearranging the tree structure.
• Traversal: Visiting all the nodes in the tree in a specific order. Common traversal methods include:
• In-order (left-root-right)
• Pre-order (root-left-right)
• Post-order (left-right-root)
• Level-order (breadth-first)

Applications of Binary Trees

Binary Search Trees (BSTs) are used for efficient searching and sorting.

Heaps are a type of binary tree used in priority queues.

Syntax Trees are used in compilers to represent the structure of expressions.

Trie (Prefix Tree) is a special type of tree used in text processing, such as auto-complete features.

Binary trees are fundamental in computer science and are used in various applications requiring hierarchical data representation and efficient data manipulation.

For Quick Faang Preparation

For solving DFS-based questions, a common approach is to use a recursive call, while for BFS (level order traversal), you typically create a queue and iterate through it. However, it's also beneficial to consider how to iterate in DFS (using a stack) and recurse in BFS-based problems.

Solving these questions will help you become familiar with basic binary tree DFS and BFS traversals.

Intuition for Level Order Traversal Iteratively Using a Queue

1. Construct a queue of type TreeNode (e.g., queue q).
2. Initially, push the given root node into the queue.
3. Iterate through the queue until it is empty:
   1. Store the current size of the queue (tempSize). This represents the size of the current level of the tree.
   2. Traverse this level by iterating while tempSize > 0:
      • Pop the current element.
      • Apply the needed operation to this element.
      • If the left or right child exists, add them to the queue.

Basic Binary Tree Problems to Enhance Your Thinking

• Binary Search Tree: Utilize the BST property judiciously. The left subtree always contains nodes with values less than the root's value, and the right subtree contains nodes with values greater than the root's value.
  Watch Our Youtube lecture on BST Concept:-
  1. Leetcode 700. Search in a Binary Search Tree
  2. Leetcode 653. Two Sum IV - Input is a BST
  3. BST Code Optimizations & Approaches (Recursive, Iterative, BFS, DFS)
• Path Problems: Given a root, perform operations on a path from the root to a leaf. Consider the type of traversal to apply when moving from root to leaf.
• Traversal-Based Tree Construction: Given combinations of preorder, postorder, and inorder traversals, construct a binary tree/BST. Hint: Observe the position of the root and the heads of the right and left subtrees in each traversal method.
• View Problems: Think about different perspectives such as left, bottom, and top views of the tree.
• Lowest Common Ancestor Problems: Given two nodes, return their ancestor at the least depth possible. These problems are essential to practice.
• Tree Validation: Miscellaneous problems to validate tree properties and structures.