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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.
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 ...
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 upheap or downheap operations can then be stated in terms of an array as follows: suppose that the heap property holds for the indices b, b+1, ..., e. The sift-down function extends the heap property to b−1, b, b+1, ..., e. Only index i = b−1 can violate the heap property.
To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it). All operations that modify the tree must adjust this information to preserve the invariant that size[x] = size[left[x]] + size[right[x]] + 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.
This implementation is a hybrid between the basic bitmap index (without compression) and the list of Row Identifiers (RID-list). Overall, the index is organized as a B+tree. When the column cardinality is low, each leaf node of the B-tree would contain long list of RIDs. In this case, it requires less space to represent the RID-lists as bitmaps.
Deletion from vEB trees is the trickiest of the operations. The call Delete(T, x) that deletes a value x from a vEB tree T operates as follows: If T.min = T.max = x then x is the only element stored in the tree and we set T.min = M and T.max = −1 to indicate that the tree is empty.