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If v has only one child w, the child pointer of v’s parent that used to point to v is now made to point to w. ) Call a node s the successor of v if the key of s is the next key immediately following the key of v in the sorted order of all the keys. It can be easily seen that when v has two children, s is the leftmost descendant in the right subtree of v. To delete v in this case, s is deleted and put in v’s place. Clearly, deletion can be done in time O(h). Because all these primitive operations require time O(h), it is important that the height of the tree be kept small.

The time taken is again O(h). Deleting an item from a binary search tree is slightly more complicated. First, using the search procedure, we locate the item in the tree. Let v be the node where the item resides. If v is a leaf, we just delete it, that is the appropriate child pointer of its parent is set to the special value NIL. If v has only one child w, the child pointer of v’s parent that used to point to v is now made to point to w. ) Call a node s the successor of v if the key of s is the next key immediately following the key of v in the sorted order of all the keys.

They are also called languages and commonly denoted by the letter L when looked upon as subsets of ͚*. The function f defining L is then notated as L. A string x ʦ L iff L(x) ϭ 1. In this section, problem means decision problem, unless explicitly stated otherwise. An algorithm M is said to decide L if M when given input x returns 1 [in symbols, M(x) ϭ 1] if x ʦ L and 0 [in symbols, M(x) ϭ 0) otherwise. The time complexity of a problem is said to be T(n) if there is an algorithm to decide it that takes no more than T(n) steps on any input of length n bits.