Proof

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/N)C:=-;

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Binary Search Trees

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The external path length is used to measure the cost of an unsuccessful search

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The insertion algorithm implies that the cost of an insert equals the cost of an unsuccessful search, which is measured by using the external path length In an insertion or unsuccessful search, we eventually reach the test NULL Recall that in a tree of N nodes there are N + 1 NULL pointers The external path length measures the total number of nodes that are accessed, including the NULL node for each of these N + I NULL pointers The NULL node is sometimes called an external tree node, which explains the term external path length As we show later in the chapter, replacing the NULL node with a sentinel may be convenient

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DEFINITION: The external path length of a binary search tree is the sum of the depths of the N + I NULL pointers The terminating NULL node is considered a node for these purposes

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One plus the result of dividing the average external path length by N + 1 yields the average cost of an unsuccessful search or insertion As with the binary search algorithm, the average cost of an unsuccessful search is only slightly more than the cost of a successful search, which follows from Theorem 192

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Theorem 192

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For any tree 7: let IPL(T) be the internal path length of T and let EPL(T) be its external path length Then, if T has N nodes, EPL(T) = IPL(T) + 2N

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Proof

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This theorem is proved by induction and is left as Exercise 199

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Random remove operations do not preserve the randomness of a tree The effects are not completely understood theoretically, but they apparently are negligible in practice

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It is tempting to say immediately that these results imply that the average running time of all operations is O(1og N) This implication is true in practice, but it has not been established analytically because the assumption used to prove the previous results do not take into account the deletion algorithm In fact, close examination suggests that we might be in trouble with our deletion algorithm because the remove operation always replaces a two-child deleted node with a node from the right subtree This result would seem to have the effect of eventually unbalancing the tree and tending to make it left-heavy It has been shown that if we build a random binary search tree and then perform roughly N2 pairs of random insertlremove combinations, the binary search trees will have an expected depth of ~ ( f i ) However, a reasonable number of random insert and remove operations (in which the order of insert and remove is also random) does not unbalance the tree in any observable way In fact, for small search trees, the remove algorithm seems to balance the tree Consequently, we can reasonably assume that for random input all

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operations behave in logarithmic average time, although this result has not been proved mathematically In Exercise 1928 we describe some alternative deletion strategies The most important problem is not the potential imbalance caused by the remove algorithm Rather, it is that, if the input sequence is sorted, the worst-case tree occurs When that happens, we are in deep trouble: We have linear time per operation (for a series of N operations) rather than logarithmic cost per operation This case is analogous to passing items to quicksort but having an insertion sort executed instead The resulting running time is completely unacceptable Moreover, it is not just sorted input that is problematic, but also any input that contains long sequences of nonrandomness One solution to this problem is to insist on an extra 5tructural condition called halarzce: No node is allowed to get too deep Any of several algorithms can be used to implement a balanced binary search tree, which has an added structure property that guarantees logarithmic depth in the worst case Most of these algorithms are much more complicated than those for the standard binary search trees, and all take longer on average for insertion and deletion They do, however, provide protection against the embarrassingly simple cases that lead to poor performance for (unbalanced) binary search trees Also, because they are balanced, they tend to give faster access time than those for the standard tres Typically, their internal path lengths are very close to the optimal N log N rather than 138N log N, so searching time is roughly 25 percent faster

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A balanced binary search tree has an added structure property to guarantee logarithmic depth in the worst case Updates are slower, but accesses are faster

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