Add the following numbers sequentially to an initially empty heap-Max: 12, 15, 9, 22, 13, 10, 7, 30, 4

To add the given numbers sequentially to an initially empty max-heap, we will follow the heap insertion algorithm for each number. The max-heap property dictates that for every node i other than the root, the value of i is less than or equal to the value of its parent. The root is the maximum element in the heap.

Here are the steps we will follow for each insertion:

1. Add the element to the bottom level of the heap at the leftmost open space.
2. Compare the added element with its parent; if it is greater, swap them.
3. Continue this process until the max-heap property is maintained (i.e., the added element is less than or equal to its parent, or it becomes the root).

Let's go through the insertions one by one:

Initial Heap

Initially, the heap is empty.

Insert 12

Inserting the first element, 12, it becomes the root as the heap was empty.

   12

Insert 15

Insert 15, and then compare and swap with 12 since 15 is greater.

   15
  /
 12

Insert 9

Insert 9. No swaps are needed since 9 is less than 15.

   15
  /  \
 12   9

Insert 22

Insert 22, and then compare and swap with 15 since 22 is greater. Then, compare 22 with 12, no swap needed.

   22
  /  \
 15   9
 /
12

Insert 13

Insert 13. No swaps are needed since 13 is less than 22 and less than 15.

   22
  /  \
 15   9
 / \
12  13

Insert 10

Insert 10. No swaps are needed since 10 is less than 22 and less than 15.

   22
  /  \
 15   9
 / \  /
12  13 10

Insert 7

Insert 7. No swaps are needed since 7 is less than 22, less than 15, and less than 9.

   22
  /  \
 15   9
 / \  / \
12  13 10  7

Insert 30

Insert 30, and then compare and swap with 22 since 30 is greater. Then, compare 30 with 15, no swap needed.

   30
  /  \
 22   9
 / \  / \
12  13 10  7

Insert 4

Insert 4. No swaps are needed since 4 is less than 30, less than 22, and less than 12.

   30
  /  \
 22   9
 / \  / \
12  13 10  7
 /
4

After all insertions, the max-heap is as follows:

       30
      /  \
    22    9
   / \   / \
  12 13 10  7
 /
4

This is the final max-heap after inserting all the given numbers sequentially. Each insertion ensures that the max-heap property is maintained.