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

Time Complexity: Kth Smallest Element in BST

O(h + k)

Understanding Time Complexity

We want to understand how the time needed to find the kth smallest element in a Binary Search Tree (BST) changes as the tree grows.

Specifically, how does the number of steps grow when the tree has more nodes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int inorder(TreeNode* root, int k, int* count, int* result) {
    if (!root) return 0;
    if (inorder(root->left, k, count, result)) return 1;
    (*count)++;
    if (*count == k) {
        *result = root->val;
        return 1;
    }
    return inorder(root->right, k, count, result);
}

int kthSmallest(TreeNode* root, int k) {
    int count = 0;
    int result = -1;
    inorder(root, k, &count, &result);
    return result;
}
    

This code performs an inorder traversal of the BST to find the kth smallest element by counting nodes visited.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive inorder traversal visiting nodes.
• How many times: Up to k nodes are visited before stopping.

How Execution Grows With Input

The number of nodes visited grows with k, not necessarily the total number of nodes n.

Input Size (n)	Approx. Operations (visits)
10	Up to k (≤ 10)
100	Up to k (≤ 100)
1000	Up to k (≤ 1000)

Pattern observation: The work depends mostly on k, the position of the element we want, not the total size n.

Final Time Complexity

Time Complexity: O(h + k)

This means the time grows with the height of the tree plus the kth position we want to find.

Common Mistake

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

[OK] Correct: The traversal stops as soon as the kth smallest is found, so often fewer than n nodes are visited, especially if k is small.

Interview Connect

Understanding how traversal limits work helps you explain efficient search in trees, a common skill interviewers look for.

Self-Check

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

Practice

1. What does the term kth smallest element in a BST mean?

easy

A. The element that appears in the kth position when BST elements are sorted in ascending order

B. The element with the kth largest value in the BST

C. The element at the kth level of the BST

D. The element with the kth highest frequency in the BST

Solution

1. Step 1: Understand BST property

   A Binary Search Tree (BST) stores elements so that left subtree has smaller values and right subtree has larger values.

2. Step 2: Define kth smallest element

   The kth smallest element is the one that appears in the kth position when all elements are sorted in ascending order.

3. Final Answer:

   The element that appears in the kth position when BST elements are sorted in ascending order -> Option A

4. Quick Check:

   kth smallest = kth in sorted order [OK]

Hint: Think of inorder traversal to get sorted order [OK]

Common Mistakes:

• Confusing kth smallest with kth largest
• Thinking kth smallest means kth level node
• Assuming frequency matters

2. Which of the following C function signatures is correct to find the kth smallest element in a BST node structure struct Node?

easy

A. void kthSmallest(int root, int k);

B. int kthSmallest(struct Node root, int k);

C. int kthSmallest(struct Node* root, int k);

D. int kthSmallest(struct Node* root);

Solution

1. Step 1: Understand function parameters

   The function needs the root pointer to the BST and the integer k to find kth smallest.

2. Step 2: Check parameter types

   Passing struct Node* for root and int for k is correct. Return type should be int for the element value.

3. Final Answer:

   int kthSmallest(struct Node* root, int k); -> Option C

4. Quick Check:

   Correct signature includes pointer and k parameter [OK]

Hint: Use pointer for tree root and int for k [OK]

Common Mistakes:

• Passing root by value instead of pointer
• Missing k parameter
• Wrong return type or missing return

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

      5
     / \
    3   7
   / \   \
  2   4   8

medium

A. 5

B. 4

C. 3

D. 7

Solution

1. Step 1: Perform inorder traversal

   Inorder traversal of BST visits nodes in ascending order: 2, 3, 4, 5, 7, 8.

2. Step 2: Identify 3rd element in traversal

   The 3rd element is 4.

3. Final Answer:

   4 -> Option B

4. Quick Check:

   Inorder 3rd element = 4 [OK]

Hint: Inorder traversal gives sorted order; pick kth element [OK]

Common Mistakes:

• Counting nodes incorrectly in traversal
• Confusing preorder or postorder with inorder
• Choosing root or other node instead of kth

4. Consider this buggy C code snippet to find kth smallest element in BST:

int count = 0;
int kthSmallest(struct Node* root, int k) {
    if (!root) return -1;
    int left = kthSmallest(root->left, k);
    if (left != -1) return left;
    count++;
    if (count == k) return root->data;
    return kthSmallest(root->right, k);
}

What is the main issue with this code?

medium

A. Global variable 'count' causes incorrect results on multiple calls

B. Function does not handle empty tree

C. Returns -1 for valid kth element

D. Does not traverse right subtree

Solution

1. Step 1: Analyze use of 'count'

   The variable 'count' is global and not reset between calls, causing wrong counts if function called multiple times.

2. Step 2: Understand impact on results

   Because 'count' is not reset, subsequent calls may return wrong kth smallest element or fail.

3. Final Answer:

   Global variable 'count' causes incorrect results on multiple calls -> Option A

4. Quick Check:

   Global count not reset breaks repeated calls [OK]

Hint: Avoid global counters; pass as function argument [OK]

Common Mistakes:

• Ignoring global variable side effects
• Assuming function works for multiple calls
• Not resetting count before each call

5. You have a BST with nodes containing duplicate values. How would you modify the standard kth smallest element algorithm to correctly find the kth smallest unique element?

hard

A. Use preorder traversal instead of inorder

B. Count all nodes including duplicates normally

C. Remove duplicates from BST before traversal

D. Use inorder traversal and skip duplicates by comparing with previous visited value

Solution

1. Step 1: Understand duplicates in BST

   Duplicates appear as multiple nodes with same value; inorder traversal visits them in sorted order including duplicates.

2. Step 2: Modify traversal to skip duplicates

   Track previous visited value during inorder traversal; only increment count when current value differs from previous.

3. Step 3: Find kth unique element

   Increment count only for unique values; when count equals k, return current node's value.

4. Final Answer:

   Use inorder traversal and skip duplicates by comparing with previous visited value -> Option D

5. Quick Check:

   Skip duplicates by comparing previous value [OK]

Hint: Track previous value to count unique elements [OK]

Common Mistakes:

• Counting duplicates as separate elements
• Using preorder traversal which is not sorted
• Trying to remove duplicates instead of skipping