Kth Smallest Element in BST in DSA Typescript - Time & Space Complexity

Time Complexity: Kth Smallest Element in BST

O(h + k)

Understanding Time Complexity

We want to know how long it takes to find the kth smallest value in a Binary Search Tree (BST).

This helps us understand how the search time grows as the tree or k grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function kthSmallest(root: TreeNode | null, k: number): number {
  let count = 0;
  let result = -1;

  function inorder(node: TreeNode | null) {
    if (!node || result !== -1) return;
    inorder(node.left);
    count++;
    if (count === k) {
      result = node.val;
      return;
    }
    inorder(node.right);
  }

  inorder(root);
  return result;
}
    

This code does an in-order traversal of the BST to find the kth smallest element.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive in-order traversal visiting nodes.
• How many times: Visits nodes until the kth smallest is found, up to k times in best case, up to n times in worst case.

How Execution Grows With Input

As the tree size (n) grows, the number of nodes visited depends on k and tree shape.

Input Size (n)	Approx. Operations (nodes visited)
10	Up to 10 (if k=10), or fewer if k smaller
100	Up to 100 (if k=100), or fewer if k smaller
1000	Up to 1000 (if k=1000), or fewer if k smaller

Pattern observation: The work grows roughly with k, but cannot exceed n (total nodes).

Final Time Complexity

Time Complexity: O(h + k)

This means the time depends on the tree height (h) plus how many nodes we visit until kth smallest.

Common Mistake

[X] Wrong: "The time is always O(n) because we might visit all nodes."

[OK] Correct: Actually, we stop once we find the kth smallest, so if k is small, we visit fewer nodes, making it faster than visiting all.

Interview Connect

Understanding this time complexity shows you can analyze recursive tree traversals and how stopping early affects performance.

Self-Check

"What if the BST is balanced vs completely skewed? How would the time complexity change?"

Practice

1. What does the "Kth Smallest Element in BST" problem ask you to find?

easy

A. The element that is the Kth smallest value in the Binary Search Tree

B. The largest element in the Binary Search Tree

C. The height of the Binary Search Tree

D. The number of nodes in the Binary Search Tree

Solution

1. Step 1: Understand the BST property

   A Binary Search Tree (BST) stores values so that left child nodes are smaller and right child nodes are larger than the parent.

2. Step 2: Define the Kth smallest element

   The Kth smallest element is the element that would appear in position K if all elements were sorted in ascending order.

3. Final Answer:

   The element that is the Kth smallest value in the Binary Search Tree -> Option A

4. Quick Check:

   Kth smallest means sorted order position K [OK]

Hint: Remember BST in-order traversal gives sorted order [OK]

Common Mistakes:

• Confusing Kth smallest with largest element
• Thinking it means tree height
• Counting nodes instead of values

2. Which TypeScript function signature correctly represents a function to find the Kth smallest element in a BST node structure?

easy

A. function kthSmallest(k: number, root: TreeNode): void

B. function kthSmallest(root: TreeNode | null, k: number): number

C. function kthSmallest(root: TreeNode, k: string): string

D. function kthSmallest(root: TreeNode, k: number): TreeNode

Solution

1. Step 1: Check parameter types and order

   The function should take the root node (which can be null) and a number k representing the position.

2. Step 2: Check return type

   The function returns the value (number) of the Kth smallest element, not a node or void.

3. Final Answer:

   function kthSmallest(root: TreeNode | null, k: number): number -> Option B

4. Quick Check:

   Correct parameter types and return number [OK]

Hint: Root node first, k as number, return value number [OK]

Common Mistakes:

• Swapping parameter order
• Using wrong return type
• Using string instead of number for k

3. Given the BST below and k = 3, what is the output of the function kthSmallest(root, 3)?

BST structure:
5
/ \
3 7
/ \ \
2 4 8

medium

A. 3

B. 5

C. 4

D. 7

Solution

1. Step 1: Perform in-order traversal

   In-order traversal visits nodes in ascending order: 2, 3, 4, 5, 7, 8.

2. Step 2: Identify the 3rd element

   The 3rd element in this order is 4.

3. Final Answer:

   4 -> Option C

4. Quick Check:

   In-order 3rd element = 4 [OK]

Hint: In-order traversal gives sorted nodes; pick Kth [OK]

Common Mistakes:

• Counting root as first without traversal
• Confusing left and right subtree order
• Picking value by node depth

4. Consider this TypeScript code snippet to find the Kth smallest element in a BST:

function kthSmallest(root: TreeNode | null, k: number): number {
  let count = 0;
  let result = -1;
  function inorder(node: TreeNode | null) {
    if (!node) return;
    inorder(node.left);
    count++;
    if (count === k) {
      result = node.val;
      return;
    }
    inorder(node.right);
  }
  inorder(root);
  return result;
}

What is the main problem with this code?

medium

A. The function does not stop traversal after finding the Kth element

B. The count variable is not incremented correctly

C. The result variable is initialized incorrectly

D. The function does not handle null root

Solution

1. Step 1: Analyze traversal stopping condition

   The code sets result when count equals k but does not stop further recursion, so it continues traversing unnecessarily.

2. Step 2: Understand impact of not stopping traversal

   Without stopping, the function wastes time and may overwrite result if not careful.

3. Final Answer:

   The function does not stop traversal after finding the Kth element -> Option A

4. Quick Check:

   Traversal must stop after Kth element found [OK]

Hint: Stop recursion immediately after finding Kth element [OK]

Common Mistakes:

• Assuming return stops all recursion
• Not using a flag to stop traversal
• Ignoring null root case

5. You want to find the Kth smallest element in a BST but the tree is very large and you need to optimize for multiple queries with different K values. Which approach is best?

hard

A. Perform in-order traversal each time for every K query

B. Use breadth-first search to find the Kth smallest element

C. Randomly pick nodes until Kth smallest is found

D. Store the in-order traversal result in an array once, then answer queries by indexing

Solution

1. Step 1: Consider repeated queries

   Performing in-order traversal each time is costly for large trees and many queries.

2. Step 2: Use pre-processing for efficiency

   Storing the sorted elements from one in-order traversal allows O(1) access for each Kth smallest query.

3. Step 3: Evaluate other options

   Breadth-first search does not give sorted order; random picking is inefficient and unreliable.

4. Final Answer:

   Store the in-order traversal result in an array once, then answer queries by indexing -> Option D

5. Quick Check:

   Precompute once, query fast [OK]

Hint: Precompute sorted list once for fast multiple queries [OK]

Common Mistakes:

• Recomputing traversal for each query
• Using BFS which is unsorted
• Random selection guessing