1. © 2004 Goodrich, Tamassia Binary Search Trees1 CSC 212 Lecture 18: Binary and AVL Trees

2. © 2004 Goodrich, Tamassia Binary Search Trees2 Ordered Dictionaries Keys are assumed to come from a total order. New operations: first(): first entry in the dictionary ordering last(): last entry in the dictionary ordering successor(k): entry with smallest key greater than or equal to k; increasing order predecessor(k): entry with largest key less than or equal to k; decreasing order

3. © 2004 Goodrich, Tamassia Binary Search Trees3 Binary Search (§ 8.3.3) Binary search can perform operation find(k) on a dictionary implemented by means of an array-based sequence, sorted by key similar to the high-low game at each step, the number of candidate items is halved terminates after O(log n) steps Example: find(7) 13457 8 91114161819 1 3 457891114161819 134 5 7891114161819 1345 7 891114161819 0 0 0 0 m l h m l h m l h l  m  h

4. © 2004 Goodrich, Tamassia Binary Search Trees4 Search Tables A search table is a dictionary implemented by means of a sorted sequence Store the items of the dictionary in an array-based sequence, sorted by key Performance: find takes O(log n) time, using binary search insert takes O(n) time since in the worst case we have to shift n  2 items to make room for the new item remove take O(n) time since in the worst case we have to shift n  2 items to compact the items after the removal Lookup table effective only for dictionaries of small size or for dictionaries on which searches are the most common operations

5. © 2004 Goodrich, Tamassia Binary Search Trees5 Binary Search Trees (§ 9.1) A binary search tree is a binary tree storing keys (or key-value entries) at its internal nodes and satisfying the following property: Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u)  key(v)  key(w) External nodes do not store items An inorder traversal of a binary search trees visits the keys in increasing order 6 92 418

6. © 2004 Goodrich, Tamassia Binary Search Trees6 Search (§ 9.1.1) To search for a key k, we trace a downward path starting at the root The next node visited depends on the outcome of the comparison of k with the key of the current node If we reach a leaf, the key is not found and we return null Example: find(4): Call TreeSearch(4,root) Algorithm TreeSearch(k, v) if T.isExternal (v) /* throw exception */ if k  key(v) return TreeSearch(k, T.left(v)) else if k  key(v) return v else { k  key(v) } return TreeSearch(k, T.right(v)) 6 9 2 4 1 8   

7. © 2004 Goodrich, Tamassia Binary Search Trees7 Search (§ 9.1.1) Example: find(5): Call TreeSearch(5,root) Algorithm TreeSearch(k, v) if T.isExternal (v) /* throw exception */ if k  key(v) return TreeSearch(k, T.left(v)) else if k  key(v) return v else { k  key(v) } return TreeSearch(k, T.right(v)) 6 9 2 4 1 8   

8. © 2004 Goodrich, Tamassia Binary Search Trees8 Insertion To perform operation insert(k, o), we search for key k (using TreeSearch) Assume k is not already in the tree, and let let w be the leaf reached by the search We insert k at node w and expand w into an internal node Example: insert(5, o) 6 92 418 6 9 2 4 18 5    w w

9. © 2004 Goodrich, Tamassia Binary Search Trees9 Deletion To perform operation remove( k ), we search for key k Assume key k is in the tree, and let let v be the node storing k If node v has a leaf child w, we remove v and w from the tree with operation removeExternal( w ), which removes w and its parent Example: remove(4) 6 9 2 4 18 5 v w 6 9 2 5 18  

10. © 2004 Goodrich, Tamassia Binary Search Trees10 Deletion (cont.) We consider the case where the key k to be removed is stored at a node v whose children are both internal we find the internal node w that follows v in an inorder traversal we copy key(w) into node v we remove node w and its left child z (which must be a leaf) by means of operation removeExternal( z ) Example: remove(3) 3 1 8 6 9 5 v w z 2 5 1 8 6 9 v 2

11. © 2004 Goodrich, Tamassia Binary Search Trees11 Performance Consider a dictionary with n items implemented by means of a binary search tree of height h the space used is O(n) methods find, insert and remove take O(h) time The height h is O(n) in the worst case and O(log n) in the best case

12. © 2004 Goodrich, Tamassia Binary Search Trees12 Your Turn Insert 4, 2, 6, 1, 3, 5, 7 into a tree Then remove nodes in this order: 4, 2, 6, 1, 3, 5, 7

13. © 2004 Goodrich, Tamassia Binary Search Trees13 AVL Tree Definition (§ 9.2) AVL trees are balanced. An AVL Tree is a binary search tree such that for every internal node v of T, the heights of the children of v can differ by at most 1. Example AVL tree where the heights are shown next to the nodes

14. © 2004 Goodrich, Tamassia Binary Search Trees14 Insertion in an AVL Tree Insertion is as in a binary search tree Always done by expanding an external node. Consider insert(54, 0): 44 1778 325088 4862 54 44 1778 325088 4862 before insertionafter insertion

15. © 2004 Goodrich, Tamassia Binary Search Trees15 Trinode Restructuring let (a,b,c) be an inorder listing of x, y, z perform the rotations needed to make b the topmost node of the three b=y a=z c=x T0T0 T1T1 T2T2 T3T3 b=y a=z c=x T0T0 T1T1 T2T2 T3T3 c=y b=x a=z T0T0 T1T1 T2T2 T3T3 b=x c=ya=z T0T0 T1T1 T2T2 T3T3 case 1: single rotation (a left rotation about a) case 2: double rotation (a right rotation about c, then a left rotation about a) (other two cases are symmetrical)

16. © 2004 Goodrich, Tamassia Binary Search Trees16 Insertion Example, continued 88 44 17 78 3250 48 62 2 4 1 1 2 2 3 1 54 1 T 0 T 1 T 2 T 3 x y z unbalanced......balanced T 1

17. © 2004 Goodrich, Tamassia Binary Search Trees17 Restructuring (as Single Rotations) Single Rotations:

18. © 2004 Goodrich, Tamassia Binary Search Trees18 Restructuring (as Double Rotations) double rotations:

19. © 2004 Goodrich, Tamassia Binary Search Trees19 Removal in an AVL Tree Removal begins as in a binary search tree, which means the node removed will become an empty external node. Its parent, w, may cause an imbalance. Remove(32): 44 17 783250 88 48 62 54 44 17 7850 88 48 62 54 before deletion of 32after deletion

20. © 2004 Goodrich, Tamassia Binary Search Trees20 Rebalancing after a Removal Let z be the first unbalanced node encountered while travelling up tree after removing a node. Let y be the child of z with the larger height, and x be the child of y with the larger height. restructure(x) restores balance at z. Restructuring may unbalance node higher in the tree, so must continue checking for balance (and restructuring where needed) until root is reached 44 17 7850 88 48 62 54 w c=x b=y a=z 44 17 78 5088 48 62 54

21. © 2004 Goodrich, Tamassia Binary Search Trees21 Running Times for AVL Trees Single restructure is O(1) Work is constant for height of tree find is O(log n) height of tree is O(log n) insert is _____________ initial find is O(log n) Restructuring up the tree – could do at most _______________________ work remove is _______________ initial find is O(log n) Restructuring up the tree – could do at most _______________________ work

22. © 2004 Goodrich, Tamassia Binary Search Trees22 Your Turn Insert 1, 2, 3, 4, 5, 6, 7, 8 into an AVL tree Then remove 1, 2, 3, 4, 5, 6, 7, 8 into AVL tree

23. © 2004 Goodrich, Tamassia Binary Search Trees23 Daily Quiz I have a “friend” who claims that the order nodes are entered into an AVL tree does not matter, the end tree is always the same Provide a small example proving my “friend” wrong