Looking up a value in a B-Tree is relatively simple and is left as an exercise for the reader. Inserting and deleting values are where all the action is. Alan Tharp [7] provides a great discussion of both inserting and deleting values in a B-Tree. In this text we provide new examples and suggest both iterative and recursive implementations of both operations.

B-Tree Insert

Inserting an item in a B-Tree involves finding the leaf node which should contain the item. It may also involve splitting if no room is left in the leaf node. When a leaf node reaches its capacity, which is two times its degree and a new item is being inserted, the 2*degree+1 items are sorted and the median value (i.e. the middle value) is promoted up the tree to the parent node. In this way, splitting may cascade up the tree.

To see the splitting process in action, consider building the tree given in Fig. 11.5 with the keys given in this order [10, 8, 22, 14, 12, 18, 2, 50, 15]. The first item to be inserted is the 10. When this occurs, the B-Tree is empty, consisting of one empty node. The (10,4) item is added into that node as shown in Fig. 11.7.

The items with keys 8, 14, and 22 are inserted in a similar fashion as shown in Fig. 11.8. The node is now full. The next item to be inserted will cause a split.

The next item inserted is a 12 causing the node to split into two nodes. The left subtree node is the original node. The right subtree contains the new node. The middle value, 12 in this case, is promoted up to the parent. In this case, there is no parent since we split the root node. In this special case a new root node is created to hold the promoted value. After taking these steps, the tree appears as shown in Fig. 11.9. The three values 18, 2, and 50 are inserted resulting in the tree as shown in

Fig. 11.10.

When 15 is inserted B-Tree node number 2 is going to split and promote the middle value, 18 in this case, up to the parent. This time there is room in the parent so the new item is added resulting in the tree shown in Fig. 11.11.

Fig. 11.7 Inserting 10 into an empty B-Tree

Fig. 11.8 After Inserting 8, 14, and 22

Fig. 11.9 After Splitting as a Result of Inserting 12

Fig. 11.10 After Inserting 18, 2, and 50

Fig. 11.11 Inserting 15 into the B-Tree Causes Splitting

Inserting an item causes one of two possible outcomes. Either the leaf node has room in it to add the new item or the leaf node splits resulting in a middle value and a new node being promoted to the parent. This suggests a recursive implementation is appropriate for inserting a new item. The recursive algorithm is given an item to insert and returns two values, the promoted key and the new right node if there is one and proceeds as follows.

1. If this is a leaf node and there is room for it, make room and store the item in the node.

2. Otherwise. if this is a leaf node, make a new node. Sort the new item and old items. Choose the middle item to promote to the parent. Take the items after the middle and put them into the new node. Return a tuple of the middle item and new right node.

3. If this is a non-leaf node, call insert recursively on the appropriate subtree. Consult the return value of the recursive call to see if there is a newly promoted key and right subtree. If so, take the appropriate action to store the new item and subtree pointer in the node. If there is no room to store the promoted value, split again as described in step 2.

Step 3 above automatically handles any cascading splits that must occur. After the recursive call the algorithm looks for any promoted value and handles it by either adding it into the node or by splitting again. An iterative version of insert would proceed in a similar manner as the recursive version except that the path to the newly inserted item would have to be maintained on a stack. Then, after inserting or splitting the leaf node, the stack of nodes on the path to the leaf would be popped one at a time, handling any promoted values, until the stack was emptied.

When writing insert as a recursive function it makes sense to implement it as a method of a B-Tree node class. Then the insert method on a B-Tree class can call the recursive insert on the B-Tree node class. In this way, if the root node is split, the B-Tree insert method can deal with this by creating a new root node from the promoted value and the left and right subtrees. Recall that the old root is the new left subtree in the newly created node.

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