Chapter 22: DSA Binary Trees
DSA Binary Trees – Let’s understand this topic properly and in depth, as if I’m sitting next to you with a whiteboard, drawing trees step by step, giving many examples, showing code patterns, explaining traversals visually, and telling you exactly why binary trees are one of the most important topics in Data Structures & Algorithms.
What is a Binary Tree?
A Binary Tree is a tree data structure in which each node has at most two children, usually called left child and right child.
Key points to remember:
- It is a hierarchical structure
- It has exactly one root node (no parent)
- Every node except root has exactly one parent
- There are no cycles
- Each node can have 0, 1, or 2 children
Visual Example – Simple Binary Tree
text
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0 1 2 3 4 5 6 7 8 9 10 11 12 |
1 ← root / \ 2 3 / \ \ 4 5 6 / / 7 8 |
This is a binary tree with:
- Root = 1
- Left subtree of root = nodes 2,4,5,7
- Right subtree of root = nodes 3,6,8
- Leaf nodes = 7,5,8 (nodes with no children)
Binary Tree Terminology (must know these perfectly)
| Term | Meaning / Explanation |
|---|---|
| Root | The topmost node (no parent) |
| Parent | Node that has children |
| Child | Node that has a parent |
| Left child / Right child | The two possible children in binary tree |
| Leaf node | Node with zero children |
| Internal node | Node with at least one child |
| Depth of a node | Number of edges from root to that node |
| Height of a node | Number of edges on the longest path from that node to a leaf |
| Height of tree | Height of root node |
| Level | All nodes at same depth (root = level 0) |
| Subtree | Tree formed by any node and all its descendants |
| Sibling | Nodes that share the same parent |
Types of Binary Trees (very frequently asked in interviews)
| Type | Definition / Property | Importance in interviews |
|---|---|---|
| Binary Tree | Each node ≤ 2 children | ★★★★★ (base structure) |
| Strict / Full Binary Tree | Every node has 0 or 2 children (no node with 1 child) | ★★★☆☆ |
| Complete Binary Tree | All levels filled except possibly last, last level filled from left to right | ★★★★★ (used in heaps) |
| Perfect Binary Tree | All internal nodes have 2 children + all leaves at same level | ★★★★☆ |
| Degenerate / Skewed Tree | Each node has only one child (looks like linked list) | ★★★☆☆ (worst case) |
| Balanced Binary Tree | Height difference between left & right ≤ 1 (or similar) | ★★★★☆ (AVL, Red-Black) |
| Binary Search Tree (BST) | Left subtree < node < right subtree (for every node) | ★★★★★ (very important) |
Binary Tree Node Structure (most common interview style)
Python
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0 1 2 3 4 5 6 7 8 9 10 |
class TreeNode: def __init__(self, val=0, left=None, right=None): self.val = val # data / value self.left = left # left child self.right = right # right child |
Most Important Binary Tree Operations / Concepts
- Construction (creating tree manually or from array/list)
- Traversal (visiting every node exactly once)
- Height / Depth / Size calculation
- Level order traversal (BFS style)
- Check properties (is BST? is balanced? is mirror?)
- Search / Insert / Delete (especially in BST)
The Four Classic Traversals (must master all)
| Traversal | Order of visiting nodes | Recursive Pattern | Typical Use Case |
|---|---|---|---|
| Pre-order | Root → Left → Right | Root first | Copy tree, prefix expression |
| In-order | Left → Root → Right | Root in middle | Sorted order in BST |
| Post-order | Left → Right → Root | Root last | Delete tree, postfix expression |
| Level-order | Level by level (left to right) | BFS using queue | Print levels, BFS applications |
Example Tree (we will use this for all traversals)
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0 1 2 3 4 5 6 7 8 9 10 |
1 / \ 2 3 / \ \ 4 5 6 |
Pre-order: 1 2 4 5 3 6 In-order: 4 2 5 1 3 6 Post-order: 4 5 2 6 3 1 Level-order: 1 2 3 4 5 6
Recursive Traversal Code (most common interview style)
Python
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 |
def preorder(root): if not root: return print(root.val, end=" ") # visit root preorder(root.left) preorder(root.right) def inorder(root): if not root: return inorder(root.left) print(root.val, end=" ") # visit root inorder(root.right) def postorder(root): if not root: return postorder(root.left) postorder(root.right) print(root.val, end=" ") # visit root |
Level Order Traversal (using Queue – very important)
Python
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 |
from collections import deque def level_order(root): if not root: return queue = deque([root]) while queue: node = queue.popleft() print(node.val, end=" ") if node.left: queue.append(node.left) if node.right: queue.append(node.right) |
Why Binary Trees are Extremely Important in Interviews
| Reason | How often asked |
|---|---|
| Fundamental hierarchical structure | ★★★★★ |
| Basis for BST, Heap, Segment Tree, Trie | ★★★★★ |
| Traversal questions appear in almost every company | ★★★★★ |
| Recursive thinking practice | ★★★★★ |
| Basis for many medium/hard tree problems | ★★★★★ |
| Used in system design (file system, DOM) | ★★★★☆ |
Most Common Binary Tree Interview Questions (sorted by frequency)
Easy → Medium level:
- Implement inorder / preorder / postorder traversal (recursive + iterative)
- Level order traversal (BFS)
- Height / maximum depth of binary tree
- Check if two trees are identical / mirror images
- Diameter of binary tree
- Invert / Mirror a binary tree
Medium → Hard:
- Validate Binary Search Tree
- Kth smallest element in BST
- Lowest Common Ancestor (LCA) in Binary Tree / BST
- Binary Tree Maximum Path Sum
- Serialize and Deserialize Binary Tree
- Construct Binary Tree from Inorder + Preorder
- Populating Next Right Pointers in Each Node
- Vertical Order Traversal / Top View / Bottom View
Quick Summary – Binary Tree Cheat Sheet
| Property / Question | Answer / Key Point |
|---|---|
| Max children per node | 2 |
| Number of edges | n-1 (for n nodes) |
| Number of leaf nodes (full binary tree) | (n+1)/2 |
| Time complexity of traversal | O(n) – visit each node once |
| Space complexity (recursive) | O(h) – recursion stack (h = height) |
| Worst case height | O(n) – skewed tree |
| Best case height | O(log n) – balanced tree |
Would you like to go deeper into any specific binary tree topic next?
Very popular next steps:
- All four traversals – recursive + iterative versions with dry runs
- Level order traversal line by line (very common question)
- Height vs Diameter vs Balanced difference
- Binary Search Tree – insert, search, delete, validation
- Lowest Common Ancestor (very frequent)
- Morris Traversal (O(1) space inorder)
Just tell me which one you want next — I’ll explain it in the same detailed, step-by-step, teacher style with examples and visuals! 😊
