BTree File Structure A BTree is a balanced

B+-Tree File Structure A B+-Tree is a balanced tree. It is a multiple-level index structure. A node of a B+-Tree of order n contains up to (n-1) search key values, sorted in order, and n pointers. Node structure of a B+-Tree of order n: P 1 K 1 P 2 . . Pn-1 Kn-1 Pn In a leaf node there at least ceil((n-1)/2) key values. Key values in the leaf nodes form a total index for the file. In the root of the tree, there at least 2 pointers. In each intermediate node, there at least ceil(n/2) pointers.

In the leaf nodes, the pointer Pi (1 <= i <= (n-1)) points to the block of the prime data area where the record with the search key value of Ki is stored. The pointer Pn is used to chain leaf nodes together to facilitate sequential searching of the file in search key order. The structure of the non-leaf nodes ( the root and the intermediate nodes) are the same as that of the leaf nodes, except all pointers point to tree nodes. The pointer Pi (2 <= i <= (n-1)) points to the subtree that contains search key values less than Ki and greater than or equal to Ki-1. The pointer P 1 points to the subtree that contains search key values less than K 1. The pointer Pn points to the subtree that contains search key values greater than or equal to Kn-1.

40 30 10 20 50 30 40 To 10 Figure 1 - A B+-Tree of Order 3 50 60

40 30 10 20 50 30 40 50 To 10 Figure 2 – Example of the Search Algorithm (for Search Key Value of 20) – STEP 1 Red arrows define the search path. Note that each search begins at the root of the tree and terminates at a leaf node. 60

40 30 10 20 50 30 40 50 To 10 Figure 2 – Example of the Search Algorithm (for Search Key Value of 20) – STEP 2 Red arrows define the search path. Note that each search begins at the root of the tree and terminates at a leaf node. 60

40 30 10 20 50 30 40 50 To 10 Figure 2 – Example of the Search Algorithm (for Search Key Value of 20) – STEP 3 (Successful search) Red arrows define the search path. Note that each search begins at the root of the tree and terminates at a leaf node. 60

40 30 10 20 50 30 35 40 Figure 3 – The Insertion Algorithm Demonstrated (Insert 35) Insertion into a leaf node with available search key space. 50 60

40 30 10 20 50 30 40 50 60 New node Figure 3 – The Insertion Algorithm Demonstrated (Insert 15) Insertion that causes node splitting. 1. Insert the record with search key value of 15 into the prime data area. 2. Search the tree for the target leaf node. 3. If there is no space in the target leaf node, acquire a new node from the operating system.

40 20 10 15 30 30 50 40 50 60 20 Figure 3 – The Insertion Algorithm Demonstrated (Insert 15) Insertion that causes node splitting. 4. Insert 15 into the target leaf and the overflowing key value of 20 into the new node. 5. Insert the overflowing search key value of 20 into the parent of the target node. If no space in the parent node, proceed recursively until either such an insertion is possible, or a new root has to be created, adding 1 to the height of the tree. 6. Adjust the pointers.

40 30 10 20 50 30 40 50 60 removed To 10 Figure 4 - The Deletion Algorithm Demonstrated (Delete 60 from the tree of Figure 1) Deletion from a leaf node that contains enough search key values after deletion. 1. Search for 60. 2. Remove 60 from the target leaf node. 3. Delete the record with search key value of 60 from the prime data area. 4. Adjust pointers.

40 20 10 15 30 30 50 40 50 60 20 Figure 4 – The Deletion Algorithm Demonstrated (Delete 20 from the tree of Figure 3 after the insertion of 15 into the tree of Figure 1) Deletion that causes combining nodes. 1. Search for 20. 2. Delete the record with search key value of 20 from the prime data area.

40 30 50 20 removed 10 15 30 40 50 60 20 Returned to O. S. Figure 4 – The Deletion Algorithm Demonstrated (Delete 20 from the tree of Figure 3 after the insertion of 15 into the tree of Figure 1) Deletion that causes combining nodes. 3. If the target node becomes empty after deletion, return the target node to the operating system. 4. Remove 20 from the parent of the target leaf node. 5. Arrange the content of the parent node of the target leaf node. 6. Adjust the pointers.

30 10 15 50 30 50 60 Figure 5 – Deletion Algorithm Demonstrated (Delete 40 from the tree of Figure 4) The tree after this deletion is shown above. Note that two intermediate nodes are combined, and the height of the tree is decreased by 1.

40 20 10 30 15 50 30 40 50 60 20 Figure 5 – The Deletion Algorithm Demonstrated (Delete 40 from the above tree) Note that the intermediate nodes cannot be combined. In this case we re-distribute the pointers. The resulting tree is shown below. 30 20 10 15 50 30 20 30 50 60