In the previous problem, show the heap after adding the four max elements.

To answer this question, we need to assume that there is an existing heap and we are adding four new elements to it. Since the question does not provide the initial heap or the values of the four max elements, I will create an example to illustrate the process.

Let's assume we have a max heap with the following values:

     50
    /  \
  30    40
 / \   / \
20 10 15 25

In a max heap, the parent node is always greater than or equal to its child nodes. Now, let's say we want to add four new elements with the following values: 60, 70, 80, 90.

We will add these elements one by one and adjust the heap to maintain the max heap property. The process of adding an element to a heap is called "heapifying."

Here are the steps to add a new element to a max heap:

1. Add the new element to the bottom level of the heap.
2. Compare the added element with its parent; if it is greater, swap them.
3. Repeat step 2 until the new element is in the correct position or it becomes the root of the heap.

Let's go through the steps for each new element.

Adding 60:

1. Add 60 to the heap:

     50
    /  \
  30    40
 / \   / \
20 10 15 25
/
60

1. Since 60 is greater than its parent (20), we swap them:

     50
    /  \
  30    40
 / \   / \
60 10 15 25
/
20

1. Now, 60 is in the correct position as it is less than its new parent (30).

Adding 70:

1. Add 70 to the heap:

     50
    /  \
  30    40
 / \   / \
60 10 15 25
/ \
20 70

1. Since 70 is greater than its parent (10), we swap them:

     50
    /  \
  30    40
 / \   / \
60 70 15 25
/ \
20 10

1. Now, 70 is greater than its new parent (30), so we swap them:

     50
    /  \
  70    40
 / \   / \
60 30 15 25
/ \
20 10

1. 70 is in the correct position as it is less than its new parent (50).

Adding 80:

1. Add 80 to the heap:

     50
    /  \
  70    40
 / \   / \
60 30 15 25
/ \ \
20 10 80

1. Since 80 is greater than its parent (15), we swap them:

     50
    /  \
  70    40
 / \   / \
60 30 80 25
/ \ \
20 10 15

1. Now, 80 is greater than its new parent (40), so we swap them:

     50
    /  \
  70    80
 / \   / \
60 30 40 25
/ \ \
20 10 15

1. 80 is in the correct position as it is less than its new parent (50).

Adding 90:

1. Add 90 to the heap:

     50
    /  \
  70    80
 / \   / \
60 30 40 25
/ \ \ \
20 10 15 90

1. Since 90 is greater than its parent (25), we swap them:

     50
    /  \
  70    80
 / \   / \
60 30 40 90
/ \ \ \
20 10 15 25

1. Now, 90 is greater than its new parent (80), so we swap them:

     50
    /  \
  70    90
 / \   / \
60 30 40 80
/ \ \ \
20 10 15 25

1. Finally, 90 is greater than its new parent (50), so we swap them:

     90
    /  \
  70    50
 / \   / \
60 30 40 80
/ \ \ \
20 10 15 25

Now, the heap is correctly adjusted, and the max heap property is maintained after adding the four max elements. Here is the final heap:

     90
    /  \
  70    50
 / \   / \
60 30 40 80
/ \ \ \
20 10 15 25

This is the heap after adding the four max elements, following the max heap property at each step.