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A B-tree of depth n+1 can hold about U times as many items as a B-tree of depth n, but the cost of search, insert, and delete operations grows with the depth of the tree. As with any balanced tree, the cost grows much more slowly than the number of elements.
If a large proportion of the elements of the tree are deleted, then the tree will become much larger than the current size of the stored elements, and the performance of other operations will be adversely affected by the deleted elements. When this is undesirable, the following algorithm can be followed to remove a value from the 2–3–4 tree:
The operation returns min. Invoke the Delete(Q, min) operation. Return min. CleanUp(Q): Delete all the elements in list L and tree T. Starting from the first element in list L, traverse the list, deleting each node. Starting from the root of the tree T, traverse the tree using the post-order traversal algorithm, deleting each node in the tree.
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A B+tree is thus particularly useful as a database system index, where the data typically resides on disk, as it allows the B+tree to actually provide an efficient structure for housing the data itself (this is described in [11]: 238 as index structure "Alternative 1").
Throughout insertion/deletion operations, the K-D-B-tree maintains a certain set of properties: The graph is a multi-way tree. Region pages always point to child pages, and can not be empty. Point pages are the leaf nodes of the tree. Like a B-tree, the path length to the leaves of the tree is the same for all queries.
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When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree (see Proof of degree bounds, below). The number of trees is decreased in the operation delete-min, where trees are linked together. As a result of a relaxed structure, some operations can take a long time while others are done very ...