Kth Smallest Element in BST in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a Binary Search Tree (BST)?

A Binary Search Tree is a tree where each node has at most two children. The left child's value is less than the parent's value, and the right child's value is greater than the parent's value. This property helps in fast searching.

beginner

What does 'Kth smallest element in BST' mean?

It means finding the element that would appear in position K if all elements of the BST were sorted in ascending order.

beginner

Which traversal method helps to find the Kth smallest element in a BST?

Inorder traversal visits nodes in ascending order in a BST, so it helps to find the Kth smallest element by counting nodes during traversal.

intermediate

What is the time complexity of finding the Kth smallest element in a BST using inorder traversal?

The time complexity is O(H + K), where H is the height of the tree. This is because we traverse down to the leftmost node (height H) and then visit K nodes in order.

advanced

How can we optimize finding the Kth smallest element if we do many queries on the same BST?

We can store the count of nodes in each subtree at every node. Then, we can decide to go left or right based on the counts, reducing the search to O(H) time per query.

Which traversal order visits BST nodes in ascending order?

A Inorder

B Preorder

C Postorder

D Level order

Inorder traversal visits left child, then node, then right child, which results in ascending order for BST.

What is the Kth smallest element in a BST?

A The element with value K

B The element at position K in sorted order

C The Kth node visited in preorder traversal

D The largest element in the BST

The Kth smallest element is the element that appears at position K when all elements are sorted.

If a BST has height H, what is the worst-case time complexity to find the Kth smallest element using inorder traversal?

A O(H * K)

B O(K)

C O(H + K)

D O(log H)

We traverse down height H to the leftmost node and then visit K nodes, so O(H + K).

Which data can be stored in BST nodes to optimize Kth smallest queries?

A Count of nodes in subtree

B Sum of subtree values

C Maximum value in subtree

D Height of subtree

Storing count of nodes in subtree helps decide which branch to explore for Kth smallest.

What is the first step in finding the Kth smallest element using inorder traversal?

A Visit the root node

B Traverse the right subtree

C Count the total nodes

D Traverse the left subtree

In inorder traversal, we first visit the left subtree to get smaller elements.

Explain how inorder traversal helps find the Kth smallest element in a BST.

Think about the order nodes are visited in inorder traversal.

Describe one way to optimize multiple Kth smallest element queries on the same BST.

Consider adding extra information to each node.

Practice

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

easy

A. The largest element in the Binary Search Tree

B. The height of the Binary Search Tree

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

D. The number of nodes in the Binary Search Tree

Solution

1. Step 1: Understand the problem statement

   The problem asks to find the Kth smallest element, meaning the element that would appear in position K if the BST elements were sorted.

2. Step 2: Recognize BST property

   In a BST, an inorder traversal visits nodes in ascending order, so the Kth visited node is the Kth smallest.

3. Final Answer:

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

4. Quick Check:

   Kth smallest = Kth visited in inorder traversal [OK]

Hint: Inorder traversal gives sorted order; pick Kth node [OK]

Common Mistakes:

• Confusing Kth smallest with Kth largest
• Ignoring BST property and sorting separately
• Using preorder or postorder traversal instead of inorder

2. Which of the following C# method signatures correctly represents a function to find the Kth smallest element in a BST node class named TreeNode?

easy

A. public int KthSmallest(TreeNode root, int k)

B. public void KthSmallest(int root, int k)

C. public int KthSmallest(TreeNode root)

D. public int KthSmallest(int k)

Solution

1. Step 1: Identify correct parameter types

   The method needs the root node of type TreeNode and an integer k to find the Kth smallest element.

2. Step 2: Check return type and parameters

   The method returns an int (the value of the Kth smallest node) and takes TreeNode root and int k as parameters.

3. Final Answer:

   public int KthSmallest(TreeNode root, int k) -> Option A

4. Quick Check:

   Correct signature includes root node and k as parameters [OK]

Hint: Method needs root node and k as inputs, returns int [OK]

Common Mistakes:

• Using void return type instead of int
• Missing root node parameter
• Using wrong parameter types

3. Given the BST below and k = 3, what will the following C# code print?

class TreeNode {
  public int val;
  public TreeNode left, right;
  public TreeNode(int v) { val = v; }
}

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

int KthSmallest(TreeNode root, int k) {
  List<int> vals = new List<int>();
  void Inorder(TreeNode node) {
    if (node == null) return;
    Inorder(node.left);
    vals.Add(node.val);
    Inorder(node.right);
  }
  Inorder(root);
  return vals[k - 1];
}

TreeNode root = new TreeNode(5);
root.left = new TreeNode(3);
root.right = new TreeNode(7);
root.left.left = new TreeNode(2);
root.left.right = new TreeNode(4);
root.right.right = new TreeNode(8);

Console.WriteLine(KthSmallest(root, 3));

medium

A. 4

B. 3

C. 5

D. 7

Solution

1. Step 1: Perform inorder traversal of BST

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

2. Step 2: Find the 3rd element in the sorted list

   The 3rd element (index 2) is 4.

3. Final Answer:

   4 -> Option A

4. Quick Check:

   Inorder sorted list = [2,3,4,5,7,8], 3rd element = 4 [OK]

Hint: Inorder traversal sorts BST; pick element at index k-1 [OK]

Common Mistakes:

• Using preorder or postorder traversal order
• Off-by-one error in indexing k-1
• Confusing left and right subtree values

4. The following code attempts to find the Kth smallest element in a BST but fails to stop the traversal early after finding the Kth element. What is the main issue?

int KthSmallest(TreeNode root, int k) {
  int count = 0;
  int result = -1;
  void Inorder(TreeNode node) {
    if (node == null) return;
    Inorder(node.left);
    count++;
    if (count == k) {
      result = node.val;
      return;
    }
    Inorder(node.right);
  }
  Inorder(root);
  return result;
}

medium

A. The function does not handle empty trees

B. The variable count is not incremented correctly

C. The result variable is not initialized

D. The return inside Inorder does not stop the entire traversal

Solution

1. Step 1: Analyze the recursive Inorder function

   The return statement inside Inorder only exits the current call, not the entire traversal, so traversal continues after finding the Kth element.

2. Step 2: Understand why this is problematic

   Continuing traversal after setting result wastes time and space; no mechanism to stop recursion early.

3. Final Answer:

   The return inside Inorder does not stop the entire traversal -> Option D

4. Quick Check:

   Return in recursion stops only current call, not whole traversal [OK]

Hint: Return in recursion stops only one call, not full traversal [OK]

Common Mistakes:

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

5. You want to find the Kth smallest element in a BST but the tree is very large and you want to optimize for time and space. Which approach is best?

Options:
A) Use an iterative inorder traversal with a stack and stop when the Kth element is found.
B) Perform a full inorder traversal and store all elements in a list, then return the Kth element.
C) Convert the BST to a sorted array by any method, then binary search for the Kth element.
D) Use preorder traversal and count nodes until K is reached.

hard

A. Perform full inorder traversal and store all elements

B. Use iterative inorder traversal with stack, stop at Kth element

C. Convert BST to sorted array then binary search for Kth element

D. Use preorder traversal and count nodes until K is reached

Solution

1. Step 1: Evaluate full inorder traversal approach

   Storing all elements uses extra space and time, not efficient for large trees.

2. Step 2: Consider iterative inorder traversal with early stop

   This approach uses a stack to traverse nodes in order and stops immediately after finding the Kth smallest, saving time and space.

3. Step 3: Analyze other options

   Binary search requires sorted array, which needs full traversal first; preorder traversal does not guarantee sorted order.

4. Final Answer:

   Use iterative inorder traversal with stack, stop at Kth element -> Option B

5. Quick Check:

   Iterative inorder + early stop = efficient Kth smallest search [OK]

Hint: Iterative inorder with stack stops early, saves time and space [OK]

Common Mistakes:

• Using full traversal wasting memory
• Using preorder which is unsorted
• Trying binary search without sorted array