BST Insert Operation in DSA Typescript - Time & Space Complexity

Time Complexity: BST Insert Operation

O(h)

Understanding Time Complexity

We want to understand how the time to insert a new value in a Binary Search Tree (BST) changes as the tree grows.

How does the number of steps needed to insert depend on the size and shape of the tree?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function insert(root: TreeNode | null, val: number): TreeNode {
  if (root === null) return new TreeNode(val);
  if (val < root.val) {
    root.left = insert(root.left, val);
  } else {
    root.right = insert(root.right, val);
  }
  return root;
}
    

This code inserts a value into a BST by moving down the tree until it finds the right spot.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls moving down one level of the tree.
• How many times: At most once per level from root to leaf, depending on tree height.

How Execution Grows With Input

Each insert moves down the tree one step at a time until it finds a spot.

Input Size (n)	Approx. Operations (steps down tree)
10	Up to 10 steps if tree is skewed, about 3-4 if balanced
100	Up to 100 steps if skewed, about 6-7 if balanced
1000	Up to 1000 steps if skewed, about 9-10 if balanced

Pattern observation: In worst case, steps grow linearly with n; in average case, steps grow slowly with tree height (log n).

Final Time Complexity

Time Complexity: O(h) where h is the tree height

This means the time depends on how tall the tree is, which can be as small as log n or as large as n.

Common Mistake

[X] Wrong: "Insertion always takes the same time regardless of tree shape."

[OK] Correct: If the tree is very unbalanced (like a linked list), insertion takes longer because you must go through many nodes.

Interview Connect

Understanding how tree shape affects insertion time helps you explain trade-offs and choose the right data structure in real problems.

Self-Check

What if we used a balanced BST like an AVL tree? How would the time complexity change?

Practice

1. What is the main purpose of the insert operation in a Binary Search Tree (BST)?

easy

A. To add a new node while keeping the BST property intact

B. To delete a node from the BST

C. To find the maximum value in the BST

D. To balance the BST after insertion

Solution

1. Step 1: Understand BST property

   A BST keeps nodes ordered so that left child is smaller and right child is larger than the parent.

2. Step 2: Purpose of insert operation

   Insert adds a new node in the correct position to maintain this order.

3. Final Answer:

   To add a new node while keeping the BST property intact -> Option A

4. Quick Check:

   Insert = Add node maintaining order [OK]

Hint: Insert adds nodes without breaking BST order [OK]

Common Mistakes:

• Confusing insert with delete operation
• Thinking insert balances the tree automatically
• Assuming insert finds max or min values

2. Which of the following is the correct TypeScript function signature for inserting a value into a BST node?

easy

A. function insert(root: Node, value: number): void

B. function insert(value: number, root: Node): void

C. function insert(root: Node | null, value: number): Node | null

D. function insert(root: Node, value: string): Node

Solution

1. Step 1: Analyze parameter types

   Insert needs the root node (which can be null if tree is empty) and the value to insert (a number).

2. Step 2: Analyze return type

   It returns the updated root node (Node or null if empty).

3. Final Answer:

   function insert(root: Node | null, value: number): Node | null -> Option C

4. Quick Check:

   Correct params and return type [OK]

Hint: Insert returns updated root, accepts root or null [OK]

Common Mistakes:

• Swapping parameter order
• Using void return type instead of returning Node
• Using wrong value type like string

3. Given the following BST and the insert operation call insert(root, 7), what will be the in-order traversal of the BST after insertion?

Initial BST in-order: 3 -> 5 -> 8 -> 10

Code snippet:

class Node {
  constructor(public val: number, public left: Node | null = null, public right: Node | null = null) {}
}

function insert(root: Node | null, val: number): Node {
  if (!root) return new Node(val);
  if (val < root.val) root.left = insert(root.left, val);
  else root.right = insert(root.right, val);
  return root;
}

medium

A. [3, 5, 8, 7, 10]

B. [3, 5, 8, 10, 7]

C. [7, 3, 5, 8, 10]

D. [3, 5, 7, 8, 10]

Solution

1. Step 1: Locate insertion point for 7

   7 is greater than 5 but less than 8, so it goes to the left of 8.

2. Step 2: Perform in-order traversal after insertion

   In-order visits left, root, right: 3, 5, 7, 8, 10.

3. Final Answer:

   [3, 5, 7, 8, 10] -> Option D

4. Quick Check:

   In-order sorted after insert [OK]

Hint: Insert left if smaller, right if larger [OK]

Common Mistakes:

• Placing 7 incorrectly after 8
• Confusing in-order traversal order
• Ignoring recursive insert logic

4. The following TypeScript code attempts to insert a value into a BST but has a bug. What is the error?

function insert(root: Node | null, val: number): Node {
  if (root === null) {
    return new Node(val);
  }
  if (val <= root.val) {
    insert(root.left, val);
  } else {
    insert(root.right, val);
  }
  return root;
}

medium

A. The base case is missing

B. The recursive calls do not update root.left or root.right

C. The function returns void instead of Node

D. The comparison operator should be < instead of <=

Solution

1. Step 1: Check recursive calls

   The calls to insert do not assign the result back to root.left or root.right, so tree structure is not updated.

2. Step 2: Correct recursive assignment

   Should be root.left = insert(root.left, val) or root.right = insert(root.right, val) to update links.

3. Final Answer:

   The recursive calls do not update root.left or root.right -> Option B

4. Quick Check:

   Must assign recursive insert result to child [OK]

Hint: Always assign recursive insert result to child node [OK]

Common Mistakes:

• Forgetting to assign recursive call result
• Misunderstanding base case presence
• Changing comparison operator unnecessarily

5. You have a BST with nodes 10, 5, 15, 3, 7, 12, 18. You want to insert 6. Which sequence of nodes will you visit during the insert operation?

hard

A. [10, 5, 7]

B. [10, 15, 12]

C. [10, 5, 3]

D. [10, 15, 18]

Solution

1. Step 1: Start at root 10

   6 is less than 10, so move to left child 5.

2. Step 2: Compare with 5 and 7

   6 is greater than 5, move to right child 7. 6 is less than 7, so insert here.

3. Final Answer:

   [10, 5, 7] -> Option A

4. Quick Check:

   Traverse nodes comparing 6 until insertion point [OK]

Hint: Follow left/right based on comparison until null [OK]

Common Mistakes:

• Visiting wrong subtree based on comparison
• Skipping nodes in path
• Confusing insertion path with traversal order