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

Recall & Review

beginner

What is a Binary Search Tree (BST)?

A BST 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. This helps in fast searching.

beginner

What does 'Kth smallest element in BST' mean?

It means finding the element that would be at position K if all elements were sorted from smallest to largest.

beginner

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

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

intermediate

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

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

intermediate

How can you stop the inorder traversal early when finding the Kth smallest element?

Keep a count of visited nodes during traversal. When count reaches K, stop traversal and return the current node's value.

What property of BST helps in finding the Kth smallest element efficiently?

A Nodes are stored in a linked list

B All nodes have two children

C Left child < parent < right child

D Nodes have random values

The BST property that left child is less than parent and right child is greater allows inorder traversal to visit nodes in sorted order.

Which traversal order visits BST nodes in ascending order?

A Inorder

B Preorder

C Postorder

D Level order

Inorder traversal visits left subtree, then node, then right subtree, resulting in ascending order for BST.

If K=1, what is the Kth smallest element in a BST?

A The smallest element

B The largest element

C The root element

D The middle element

The 1st smallest element is the smallest element in the BST.

What is the main idea to stop traversal early when finding the Kth smallest element?

A Stop after visiting right subtree

B Stop after visiting all nodes

C Stop after visiting root

D Stop after visiting K nodes

We count nodes visited during inorder traversal and stop when count equals K.

What is the worst-case time complexity to find the Kth smallest element in a skewed BST?

A O(log N)

B O(N)

C O(1)

D O(N log N)

In a skewed BST, height H can be N, so time complexity is O(N).

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

Think about the order nodes are visited and how counting helps.

Describe the steps to implement a function that returns the Kth smallest element in a BST.

Focus on traversal and counting.

Practice

1. What does the term Kth Smallest Element in BST mean?

easy

A. The element that is the Kth largest in the BST

B. The element that is the Kth in ascending order in the BST

C. The root element of the BST

D. The element with value K in the BST

Solution

1. Step 1: Understand BST ordering

   A Binary Search Tree (BST) stores elements so that left child is smaller and right child is larger.

2. Step 2: Define Kth smallest element

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

3. Final Answer:

   The element that is the Kth in ascending order in the BST -> Option B

4. Quick Check:

   Kth smallest means sorted order position [OK]

Hint: Kth smallest means counting nodes in ascending order [OK]

Common Mistakes:

• Confusing Kth smallest with Kth largest
• Thinking it means the root node
• Assuming K is the value, not position

2. Which of the following Python function signatures is correct to find the Kth smallest element in a BST?

easy

A. def kth_smallest(root: Optional[TreeNode], k: int) -> int:

B. def kth_smallest(k: int, root: TreeNode) -> int:

C. def kth_smallest(root: TreeNode, k: str) -> int:

D. def kth_smallest(root: TreeNode) -> int:

Solution

1. Step 1: Check parameter order and types

   The function should take the root node and an integer k to find the Kth smallest element.

2. Step 2: Validate correct typing and order

   def kth_smallest(root: Optional[TreeNode], k: int) -> int: correctly uses root as TreeNode (or Optional) and k as int, returning int.

3. Final Answer:

   def kth_smallest(root: Optional[TreeNode], k: int) -> int: -> Option A

4. Quick Check:

   Correct parameter order and types [OK]

Hint: Root node first, k as int second in function parameters [OK]

Common Mistakes:

• Swapping parameter order
• Using wrong type for k (like string)
• Missing k parameter

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

       5
      / \
     3   7
    / \   \
   2   4   8

medium

A. 4

B. 5

C. 3

D. 7

Solution

1. Step 1: List BST elements in ascending order

   Inorder traversal gives: 2, 3, 4, 5, 7, 8.

2. Step 2: Identify the 3rd smallest element

   The 3rd element in this list is 4.

3. Final Answer:

   4 -> Option A

4. Quick Check:

   Inorder 3rd element = 4 [OK]

Hint: Inorder traversal lists nodes ascending; pick Kth [OK]

Common Mistakes:

• Choosing root or random node instead of Kth
• Counting nodes incorrectly
• Confusing Kth smallest with Kth largest

4. The following code tries to find the Kth smallest element in a BST but has a bug. What is the bug?

def kth_smallest(root, k):
    stack = []
    while True:
        while root:
            stack.append(root)
            root = root.right  # Bug here
        root = stack.pop()
        k -= 1
        if k == 0:
            return root.val
        root = root.left

medium

A. Stack should be initialized with root

B. Return statement is missing

C. k should be incremented, not decremented

D. It should go to the left child first, not right

Solution

1. Step 1: Analyze traversal direction

   For Kth smallest, we do inorder traversal: left -> node -> right. The code wrongly goes right first.

2. Step 2: Identify correct child to traverse first

   It should go to the left child first to get smallest elements first.

3. Final Answer:

   It should go to the left child first, not right -> Option D

4. Quick Check:

   Inorder traversal starts with left child [OK]

Hint: Inorder traversal: always left child first [OK]

Common Mistakes:

• Going right child first in inorder
• Not using stack properly
• Incorrect decrement of k

5. You have a BST with nodes: 10, 5, 15, 3, 7, 12, 18. You want to find the 5th smallest element. Which approach below correctly finds it efficiently?

hard

A. Use preorder traversal and return the 5th node visited

B. Traverse only right subtree and pick the 5th node

C. Perform inorder traversal and return the 5th visited node

D. Return the root node value directly

Solution

1. Step 1: Understand traversal types

   Inorder traversal visits nodes in ascending order in BST.

2. Step 2: Apply inorder traversal to find Kth smallest

   Perform inorder traversal and stop at the 5th node visited to get the 5th smallest element.

3. Final Answer:

   Perform inorder traversal and return the 5th visited node -> Option C

4. Quick Check:

   Inorder traversal gives sorted order [OK]

Hint: Use inorder traversal to get sorted nodes [OK]

Common Mistakes:

• Using preorder or postorder which are not sorted
• Ignoring left subtree nodes
• Assuming root is always Kth smallest