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Create a order 3 B-Tree for the following key data    10, 6, 12, 18, 13, 15, 3, 2, 1. What are the key values at the leaf nodes from left to write in the sequence.*1 point1,2,6,12,15,182,6,10,13,15,182,10,12,15,1818,15,3,2,2

Question

Create a order 3 B-Tree for the following key data    10, 6, 12, 18, 13, 15, 3, 2, 1. What are the key values at the leaf nodes from left to write in the sequence.*1 point1,2,6,12,15,182,6,10,13,15,182,10,12,15,1818,15,3,2,2

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Solution

To create a B-Tree, we start with an empty tree and insert the given key values one by one.

  1. Start with an empty B-Tree of order 3.
  2. Insert the first key value, 10, into the tree. Since the tree is empty, it becomes the root of the tree.
  3. Insert the second key value, 6, into the tree. Since the root already has space, we can directly insert it as the left child of the root.
  4. Insert the third key value, 12, into the tree. Again, the root has space, so we insert it as the right child of the root.
  5. Insert the fourth key value, 18, into the tree. The root is full now, so we need to split it. The middle key value, 12, moves up to become the new root, and the left and right children of the root become the left and right children of the new root, respectively. The key value 18 is inserted as the right child of the new root.
  6. Insert the fifth key value, 13, into the tree. We start from the root and find the appropriate child to insert the key value. In this case, it is the right child of the root. Since the right child is full, we split it in the same way as before. The middle key value, 15, moves up to become the new root, and the left and right children of the right child become the left and right children of the new root, respectively. The key value 13 is inserted as the right child of the new root.
  7. Insert the sixth key value, 15, into the tree. Again, we start from the root and find the appropriate child to insert the key value. In this case, it is the right child of the new root. Since the right child is full, we split it in the same way as before. The middle key value, 15, moves up to become the new root, and the left and right children of the right child become the left and right children of the new root, respectively. The key value 15 is inserted as the right child of the new root.
  8. Insert the seventh key value, 3, into the tree. We start from the root and find the appropriate child to insert the key value. In this case, it is the left child of the root. Since the left child is full, we split it in the same way as before. The middle key value, 6, moves up to become the new root, and the left and right children of the left child become the left and right children of the new root, respectively. The key value 3 is inserted as the right child of the new root.
  9. Insert the eighth key value, 2, into the tree. We start from the root and find the appropriate child to insert the key value. In this case, it is the left child of the new root. Since the left child is full, we split it in the same way as before. The middle key value, 3, moves up to become the new root, and the left and right children of the left child become the left and right children of the new root, respectively. The key value 2 is inserted as the right child of the new root.
  10. Insert the ninth key value, 1, into the tree. We start from the root and find the appropriate child to insert the key value. In this case, it is the left child of the new root. Since the left child is full, we split it in the same way as before. The middle key value, 2, moves up to become the new root, and the left and right children of the left child become the left and right children of the new root, respectively. The key value 1 is inserted as the right child of the new root.

The resulting B-Tree with the given key values is as follows:

                2, 6, 10, 12, 15, 18
               /    |    |    |    \
              1     3    6    13    15, 18

The key values at the leaf nodes from left to right in the sequence are: 1, 2, 6, 10, 13, 15, 18.

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