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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:
To delete the minimum element from the heap, first find this element, remove it from the root of its binomial tree, and obtain a list of its child subtrees (which are each themselves binomial trees, of distinct orders). Transform this list of subtrees into a separate binomial heap by reordering them from smallest to largest order.
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
This operation is done to determine if the element is either in the list or in a 2–4 tree. A 2–4 tree is used when a delete operation occurs. If the item x is already in tree T, the item is removed using the 2–4 tree delete operation. Otherwise, the item x is in list L (done by checking if the bit variable is set). All the elements stored ...
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.
This is a fundamental requirement for the data-structure algorithms on B-tree to work. In particular then, a 2-3 tree is not a B-tree; it's algorithms are different than those of B-trees. The text currently uses 2-3 trees as a recurring example; those examples should be changed to refer to the 2-3-4-tree, the smallest example of a B-tree.