Construct a tree of order 5 for the list of elements 2,8,5,6,13,12,9,14,18,15,16,21

To construct a tree of order 5, we will be using a B-tree data structure. A B-tree of order 5 means that any internal node (except the root) can have a minimum of 5 children and a maximum of 10 children. The root can have a minimum of 2 children and a maximum of 10 children.

Let's start constructing the B-tree for the given list of elements: 2,8,5,6,13,12,9,14,18,15,16,21.

Step 1: Start with an empty tree. Insert the first element 2 into the tree.

2

Step 2: Insert the second element 8 into the tree. Since 8 is greater than 2, it is placed to the right of 2.

2,8

Step 3: Insert the third element 5 into the tree. Since 5 is greater than 2 but less than 8, it is placed between 2 and 8.

2,5,8

Step 4: Insert the fourth element 6 into the tree. Since 6 is greater than 5 but less than 8, it is placed between 5 and 8.

2,5,6,8

Step 5: Insert the fifth element 13 into the tree. Since 13 is greater than all the elements in the tree, it is placed at the rightmost position.

2,5,6,8,13

Step 6: Insert the sixth element 12 into the tree. Since 12 is greater than 8 but less than 13, it is placed between 8 and 13. But the root node is already full (it can only contain 5 elements), so we need to split the node. The middle element 6 becomes the new root, and the elements on the left and right of 6 become the children of the new root.

      6
     / \
   2,5  8,12,13

Step 7: Insert the seventh element 9 into the tree. Since 9 is greater than 6 but less than 8, it is placed between 6 and 8 in the right child of the root.

      6
     / \
   2,5  8,9,12,13

Step 8: Insert the eighth element 14 into the tree. Since 14 is greater than all the elements in the right child of the root, it is placed at the rightmost position. But the right child is already full, so we need to split the node. The middle element 9 becomes the new root, and the elements on the left and right of 9 become the children of the new root.

        6
       / \
     2,5  9
         / \
       8  12,13,14

Step 9: Continue this process until all the elements are inserted into the tree. The final B-tree of order 5 for the given list of elements is:

            6,9
           / | \
         2,5 8  12,14
                  / | \
                13  15,16,18,21

In this B-tree, each internal node (except the root) has at least 5 children and at most 10 children, and the root has at least 2 children and at most 10 children. Therefore, it is a B-tree of order 5.