Why Floor and Ceil in BST in DSA C++?

The Big Idea

What if you could find the closest number to your target 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 to find floor or ceil means checking each number one by one. This takes a lot of time if the list is long. It's easy to make mistakes, like missing the closest number or mixing up floor and ceil.

The Solution

A Binary Search Tree (BST) organizes numbers so you can quickly jump to the part of the tree that might have the floor or ceil. This way, you don't check every number, just a few steps, making it fast and reliable.

Before vs After

✗ Before

int floor = INT_MIN;
for (int num : sortedList) {
  if (num <= target) floor = num;
}

✓ After

int floor = findFloor(root, target);

What It Enables

This lets you quickly find the closest smaller or larger number to any target, enabling fast decisions and searches in sorted data.

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.

Floor and ceil help find closest smaller or larger values quickly.

Practice

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

easy

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

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

C. The root value of the BST

D. The average of all values in the BST

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 other options

   The smallest value in the BST greater than or equal to the given value describes ceil, while the remaining options are unrelated to floor concept.

3. Final Answer:

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

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(int val);

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

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

D. int ceilBST(Node* root, int val);

Solution

1. Step 1: Analyze function parameters and return type

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

2. Step 2: Check each option

   int ceilBST(Node* root, int val); matches correct signature: returns int, takes Node* and int. Others have wrong parameter order, missing pointers, or wrong return type.

3. Final Answer:

   int ceilBST(Node* root, int val); -> Option D

4. Quick Check:

   Correct signature returns int and takes Node* root [OK]

Hint: Function needs root pointer and value, returns int [OK]

Common Mistakes:

• Passing Node by value instead of pointer
• Wrong parameter order
• Incorrect return type

3. Given the BST below and the value 14, what is the ceil value?

        10
       /  \
      5    15
          /  \
         12   20

medium

A. 12

B. 15

C. 20

D. 10

Solution

1. Step 1: Locate value 14 in BST context

   14 is not in BST. We look for smallest value ≥ 14.

2. Step 2: Check nodes greater than or equal to 14

   Nodes ≥14 are 15 and 20. The smallest among these is 15.

3. Final Answer:

   15 -> Option B

4. Quick Check:

   Ceil(14) = 15 [OK]

Hint: Ceil is smallest node ≥ target [OK]

Common Mistakes:

• Choosing 12 which is less than 14
• Choosing 20 which is larger but not smallest ≥14
• Confusing floor and ceil

4. Consider this buggy C++ code snippet to find floor in BST:

int floorBST(Node* root, int val) {
  if (!root) return -1;
  if (root->data == val) return root->data;
  if (root->data > val) return floorBST(root->left, val);
  int floorRight = floorBST(root->right, val);
  return (floorRight != -1 && floorRight <= val) ? floorRight : root->data;
}

What is the bug?

medium

A. It always returns root->data without recursion

B. It does not check for null root

C. It returns root->data when floorRight is greater than val, which is incorrect

D. It returns -1 for all values

Solution

1. Step 1: Understand the floor logic in code

   If root->data > val, search left subtree. Else search right subtree and compare.

2. Step 2: Identify incorrect return condition

   The code returns root->data if floorRight > val. However, since floorRight is never > val (returns -1 or ≤ val), this never triggers, causing it to return -1 (when right subtree has no floor ≤ val) instead of root->data.

3. Final Answer:

   It returns root->data when floorRight is greater than val, which is incorrect -> Option C

4. Quick Check:

   Floor must be ≤ val, code returns invalid value [OK]

Hint: Floor must never exceed val; check return conditions [OK]

Common Mistakes:

• Returning values greater than val as floor
• Missing null checks
• Ignoring recursion on left subtree

5. You have a BST with nodes: 8, 4, 12, 2, 6, 10, 14. You want to find both floor and ceil for value 7. Which pair is correct?

hard

A. Floor = 6, Ceil = 8

B. Floor = 8, Ceil = 6

C. Floor = 4, Ceil = 10

D. Floor = 7, Ceil = 7

Solution

1. Step 1: Identify floor of 7 in BST

   Floor is greatest value ≤7. Values ≤7 are 2,4,6. Greatest is 6.

2. Step 2: Identify ceil of 7 in BST

   Ceil is smallest value ≥7. Values ≥7 are 8,10,12,14. Smallest is 8.

3. Final Answer:

   Floor = 6, Ceil = 8 -> Option A

4. Quick Check:

   Floor ≤7 = 6, Ceil ≥7 = 8 [OK]

Hint: Floor ≤ val, Ceil ≥ val; pick closest values [OK]

Common Mistakes:

• Swapping floor and ceil values
• Assuming floor or ceil equals the value if not present
• Choosing values not in BST