Chapter 10: Search Trees

1. Chapter 10: Search Trees Objectives: Binary Search Trees: Search, update, and implementation AVL Trees: Properties and maintenance 2-4 Trees: Properties and maintenance Red-black Trees: Properties and equivalence to 2-4 Trees CSC311: Data Structures

2. 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 • successors(k): iterator of entries with keys greater than or equal to k; increasing order • predecessors(k): iterator of entries with keys less than or equal to k; decreasing order CSC311: Data Structures

3. 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 l=m =h Binary Search • 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) CSC311: Data Structures

4. A search table is a dictionary implemented by means of a sorted sequence We store the items of the dictionary in an array-based sequence, sorted by key We use an external comparator for the keys 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 The lookup table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations) Search Tables CSC311: Data Structures

5. 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 2 9 1 4 8 Binary Search Trees CSC311: Data Structures

6. 6 < 2 9 > = 8 1 4 Search AlgorithmTreeSearch(k, v) ifT.isExternal (v) returnv if k<key(v) returnTreeSearch(k, T.left(v)) else if k=key(v) returnv else{ k>key(v) } returnTreeSearch(k, T.right(v)) • 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 nukk • Example: find(4): • Call TreeSearch(4,root) CSC311: Data Structures

7. 6 < 2 9 > 1 4 8 > w 6 2 9 1 4 8 w 5 Insertion • To perform operation inser(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 CSC311: Data Structures

8. 6 < 2 9 > v 1 4 8 w 5 6 2 9 1 5 8 Deletion • To perform operation remove(k), we search for key k • Assume key k is in the tree, and 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 CSC311: Data Structures

9. 1 v 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9 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 CSC311: Data Structures

10. 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 CSC311: Data Structures

11. AVL Tree Definition • 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. An example of an AVL tree where the heights are shown next to the nodes. CSC311: Data Structures

12. Height of an AVL Tree • Fact: The height of an AVL tree storing n keys is O(log n). • Proof: Let us bound n(h): the minimum number of internal nodes of an AVL tree of height h. • We easily see that n(1) = 1 and n(2) = 2 • For n > 2, an AVL tree of height h contains the root node, one AVL subtree of height n-1 and another of height n-2. • That is, n(h) = 1 + n(h-1) + n(h-2) • Knowing n(h-1) > n(h-2), we get n(h) > 2n(h-2). So n(h) > 2n(h-2), n(h) > 4n(h-4), n(h) > 8n(n-6), … (by induction), n(h) > 2in(h-2i) • Solving the base case we get: n(h) > 2 h/2-1 • Taking logarithms: h < 2log n(h) +2 • Thus the height of an AVL tree is O(log n) CSC311: Data Structures

13. 44 17 78 44 32 50 88 17 78 48 62 32 50 88 54 48 62 Insertion in an AVL Tree • Insertion is as in a binary search tree • Always done by expanding an external node. • Example: c=z a=y b=x w before insertion after insertion CSC311: Data Structures

14. a=z a=z T0 c=y T0 b=y b=x T3 c=x T1 b=y b=x T1 T2 T2 T3 a=z c=x a=z c=y T2 T3 T1 T3 T0 T0 T2 T1 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 case 2: double rotation (a right rotation about c, then a left rotation about a) (other two cases are symmetrical) case 1: single rotation (a left rotation about a) CSC311: Data Structures

15. 44 x 3 2 17 62 z y 2 1 2 32 78 50 1 1 1 ...balanced 54 88 48 T 2 T T T T 0 1 1 3 Insertion Example, continued unbalanced... 4 CSC311: Data Structures

16. Restructuring: Single Rotations • Single Rotations: CSC311: Data Structures

17. Restructuring: Double Rotations • double rotations: CSC311: Data Structures

18. 44 17 62 32 50 78 88 48 54 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. • Example: 44 17 62 50 78 88 48 54 before deletion of 32 after deletion CSC311: Data Structures

19. 62 44 a=z 44 78 17 62 w b=y 17 50 88 50 78 c=x 48 54 88 48 54 Rebalancing after Removal • Let z be the first unbalanced node encountered while travelling up the tree from w. Also, let y be the child of z with the larger height, and let x be the child of y with the larger height. • We perform restructure(x) to restore balance at z. • As this restructuring may upset the balance of another node higher in the tree, we must continue checking for balance until the root of T is reached CSC311: Data Structures

20. Running Times for AVL Trees • a single restructure is O(1) • using a linked-structure binary tree • find is O(log n) • height of tree is O(log n), no restructures needed • insert is O(log n) • initial find is O(log n) • Restructuring up the tree, maintaining heights is O(log n) • remove is O(log n) • initial find is O(log n) • Restructuring up the tree, maintaining heights is O(log n) CSC311: Data Structures