Which is min heap for the given data 15 6 15 50 3 33 45 40 80 10

Given Data:

15 6 15 50 3 33 45 40 80 10

Min Heap:

A min heap is a complete binary tree where the value of each node is less than or equal to the values of its children. The root node contains the minimum value in the heap.

Heapify Operation:

The heapify operation is used to maintain the min heap property after insertion or deletion of an element. It recursively compares a node with its children and swaps the values if the node is greater than any of its children.

Building the Min Heap:

1. Start from the last non-leaf node (the one just above the last level).
2. Perform the heapify operation on this node.
3. Move to the next non-leaf node to its left.
4. Repeat steps 2 and 3 until you reach the root node.

Min Heap Representation of the Given Data:

        10
       /  \
      3    15
     / \    / \
    6   15  33  50
   /     /  \
  40    45   80

Explanation:

1. Start from the last non-leaf node, which is 33.
2. Heapify 33 by comparing it with its children 45 and 80. Since 33 is less than both its children, no swapping is required.
3. Move to the next non-leaf node to its left, which is 15.
4. Heapify 15 by comparing it with its children 40 and 50. Since 15 is less than both its children, no swapping is required.
5. Move to the next non-leaf node to its left, which is 3.
6. Heapify 3 by comparing it with its children 6 and 15. Since 3 is less than both its children, no swapping is required.
7. Repeat steps 2-6 until you reach the root node.

The final heap representation is shown in the diagram above, which is a min heap.

Conclusion:

The given data, when arranged in a min heap, results in the following heap representation:

        10
       /  \
      3    15
     / \    / \
    6   15  33  50
   /     /  \
  40    45   80

This heap representation satisfies the min heap property, where each node is less than or equal to its children, and the root node contains the minimum value (10 in this case).