Why Floor and Ceil in BST in DSA C?

The Big Idea

What if you could find the closest smaller or bigger number instantly without checking every single one?

The Scenario

Imagine you have a sorted list of numbers on paper, and you want to find the closest number less than or equal to a target (floor), or the closest number greater than or equal to the target (ceil). Doing this by scanning the list every time is slow and tiring.

The Problem

Manually scanning a list for floor or ceil means checking each number one by one. This is slow for large lists and easy to make mistakes, especially if the list is very long or changes often.

The Solution

A Binary Search Tree (BST) organizes numbers so you can quickly find floor and ceil by moving left or right based on comparisons. This saves time and reduces errors.

Before vs After

✗ Before

int floor = -1;
for (int i = 0; i < n; i++) {
  if (arr[i] <= target) floor = arr[i];
}

✓ After

int floor = findFloor(root, target);

What It Enables

This lets you quickly find closest smaller or larger values in sorted data, enabling fast decisions and queries.

Real Life Example

When booking a flight, you want the closest cheaper or slightly more expensive ticket than your budget. Floor and ceil in BST help find these prices fast.

Key Takeaways

Manual search is slow and error-prone for floor and ceil.

BST structure speeds up finding floor and ceil efficiently.

Useful for quick range queries and decision making.

Practice

1. What does the floor of a value in a Binary Search Tree (BST) represent?

easy

A. The smallest value in the BST greater than or equal to the given value

B. The root value of the BST

C. The greatest value in the BST less than or equal to the given value

D. The smallest value in the BST less than the given value

Solution

1. Step 1: Understand the definition of floor in BST

   The floor of a value is the greatest value in the BST that is less than or equal to the given value.

2. Step 2: Compare with options

   The greatest value in the BST less than or equal to the given value matches this definition exactly, while others describe different concepts.

3. Final Answer:

   The greatest value in the BST less than or equal to the given value -> Option C

4. Quick Check:

   Floor = greatest ≤ value [OK]

Hint: Floor is largest value ≤ target in BST [OK]

Common Mistakes:

• Confusing floor with ceil
• Thinking floor is the smallest value
• Assuming floor is always the root

2. Which of the following is the correct C function signature to find the ceil of a value in a BST node structure named Node?

easy

A. int ceilBST(Node* root, int key);

B. void ceilBST(Node root, int key);

C. Node* ceilBST(int key, Node* root);

D. int ceilBST(int key);

Solution

1. Step 1: Analyze function parameters and return type

   The function should take a pointer to the root node and the key to find ceil for, returning an integer value.

2. Step 2: Match with options

   int ceilBST(Node* root, int key); matches this: it takes Node* root and int key, returns int. Others have wrong parameter order, types, or return type.

3. Final Answer:

   int ceilBST(Node* root, int key); -> Option A

4. Quick Check:

   Correct signature: pointer root, key, returns int [OK]

Hint: Function takes root pointer and key, returns int [OK]

Common Mistakes:

• Passing root by value instead of pointer
• Wrong parameter order
• Returning Node* instead of int

3. Given the BST below and the key = 14, what is the output of the ceil function?

        10
       /  \
      5    15
          /  \
         12   20

medium

A. -1

B. 15

C. 12

D. 20

Solution

1. Step 1: Understand ceil definition and traverse BST

   Ceil is smallest value ≥ key. Starting at root 10, since 14 > 10, move right to 15. 15 ≥ 14, so candidate ceil is 15. Check left subtree of 15 (12) which is < 14, so ignore.

2. Step 2: Confirm smallest candidate

   15 is the smallest node ≥ 14. 20 is larger but not smaller than 15, so 15 is ceil.

3. Final Answer:

   15 -> Option B

4. Quick Check:

   Ceil(14) = 15 [OK]

Hint: Ceil is smallest node ≥ key, check right subtree carefully [OK]

Common Mistakes:

• Choosing 20 instead of 15
• Choosing 12 which is < 14
• Returning -1 when ceil exists

4. Consider the following buggy C code snippet to find floor in BST. What is the main error?

int floorBST(Node* root, int key) {
    if (!root) return -1;
    if (root->data == key) return root->data;
    if (root->data > key) return floorBST(root->left, key);
    int floorValue = floorBST(root->right, key);
    if (floorValue <= key && floorValue != -1) return floorValue;
    else return root->data;
}

medium

A. It returns root->data even if floorValue is -1

B. It does not check for NULL root

C. It returns -1 when key is found

D. It incorrectly searches left subtree when root->data < key

Solution

1. Step 1: Analyze return conditions

   If floorValue is -1 (meaning no floor found in right subtree), the code returns root->data without checking if root->data ≤ key, which can be incorrect.

2. Step 2: Identify the error

   The code should check if root->data ≤ key before returning it; otherwise, it may return a value greater than key, violating floor definition.

3. Final Answer:

   It returns root->data even if floorValue is -1 -> Option A

4. Quick Check:

   Missing check before returning root->data causes wrong floor [OK]

Hint: Check root->data ≤ key before returning it [OK]

Common Mistakes:

• Ignoring floorValue == -1 case
• Returning wrong subtree
• Not handling NULL root

5. You have a BST with nodes: 8, 4, 12, 2, 6, 10, 14. You want to find the floor and ceil of 5. Which pair is correct?

hard

A. Floor = 5, Ceil = 5

B. Floor = 6, Ceil = 4

C. Floor = 2, Ceil = 8

D. Floor = 4, Ceil = 6

Solution

1. Step 1: Identify floor of 5 in BST

   Floor is greatest value ≤ 5. Nodes ≤ 5 are 2 and 4. Greatest is 4.

2. Step 2: Identify ceil of 5 in BST

   Ceil is smallest value ≥ 5. Nodes ≥ 5 are 6, 8, 10, 12, 14. Smallest is 6.

3. Final Answer:

   Floor = 4, Ceil = 6 -> Option D

4. Quick Check:

   Floor(5)=4, Ceil(5)=6 [OK]

Hint: Floor ≤ key, ceil ≥ key; pick closest values [OK]

Common Mistakes:

• Swapping floor and ceil values
• Assuming floor or ceil equals key if not present
• Choosing wrong closest values