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PERFORMANCE OF BINARY SEARCH TREES Both the insert() and the search() functions begin at the root of the tree and proceed down toward the leaves, making one comparison at each level of the tree. Therefore the time required to execute either algorithm is proportional to h + 1, where h is the height of the tree. The search() function may terminate before reaching a leaf, but h + 1 is still an upper bound on the number of comparisons that it can make. Theorem 12.2 In a binary search tree of size n, the insert() and the search() functions each require O(lgn) comparisons in the best case. In the best case, the binary tree is completely balanced and nearly full, so by Corollary 11.2 on page 202, h = lg(n+1) 1 = O(lgn). Theorem 12.3 In a binary search tree of size n, the insert() and the search() functions each require O(n) comparisons in the worst case. In the worst case the tree is linear, so h + 1 = n = O(n).

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Theorem 12.4 In a binary search tree of size n, the insert() and the search() functions each require O(2lnn) O(1.39lgn) comparisons in the average case. The proof of this result is beyond the scope of this outline.

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5. If k i = k, return false (indicating that the insertion was unsuccessful because a record with key k already exists, and keys should be unique). 6. Let x be the root of subtree S i . 7. Add the record to disk. 8. Insert k (with the record s disk address) into x between k i 1 and k i . 9. Add a dummy leaf node to x. 10. If degree(x) = m, repeat steps 11 13 until degree(x) < m. 11. Let k j be the middle key in node x. 12. Let u and v be the left and right halves of x after removing k j from x. 13. If x is the root, create a new root node containing k j with subtrees u and v. 14. Otherwise, insert k j in x s parent node and attach subtrees u and v. 15. Return true. This insertion process is illustrated in Figure 12.6.

Suppose that this system is implemented on a 16-bit fixed-point processor and that the sums of products a are accumulated prior to quantization. Let : be the variance of the round-off noise.

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(a) If the system is implemented in direct form 11, find the variance of the round-off noise at the output of the filter.

CHAP. 12]

The deletion algorithm for B-trees is similar to the insertion algorithm. All three algorithms run in time proportional to the height of the tree. From Corollary 10.1 on page 188 it follows that that height is proportional to log m n. From Theorem A.2 on page 320, it follows that that is proportional to lgn. Thus we have: Theorem 12.1 In a B-tree, searching, inserting, and deleting all run in O(lgn) time. BINARY SEARCH TREES A binary search tree is a binary tree whose elements include a key field of some ordinal type and which has this property: If k is the key value at any node, then k x for every key x in the node s left subtree and k y for

( a ) The direct form I1 implementation of this system is shown in the figure below along with the two round-off noise sources.

CHAP. 12]

AVL TREES The imbalance problem illustrated in Example 12.6 can be avoided by imposing balance constraints on the nodes of the binary search tree. Define the balance number at any node of a tree to be the difference between the height of its left subtree and the height of its right subtree. An AVL tree is a binary search tree where the balance number at each node is either 1, 0, or 1. The name comes from the two inventors of this method: G.M. Adel son-Velskii and Y.M. Landis. The tree in Figure 12.12 is not an AVL tree because it is imbalanced at node C. Its balance number there is 2, which is outside the allowable range. It is also imbalanced at node G. The tree in Figure 12.11 is an AVL tree: Every balance number is either 1, 0 or 1. EXAMPLE 12.7 An AVLTree CLASS

Because the sum x(n)

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