7.3 AVL-Trees. Definition 15. Lemma 16. AVL-trees are binary search trees that fulfill the following balance condition.

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1 Definition 15 AVL-trees are binary search trees that fulfill the following balance condition. F eery node height(left sub-tree()) height(right sub-tree()) 1. Lemma 16 An AVL-tree of height h contains at least F h+2 1 and at most 2 internal nodes, where F n is the n-th Fibonacci number (F 0 = 0, F 1 = 1), and the height is the maimal number of edges from the root to an (empty) dummy leaf. c Ernst Mayr, Harald Räcke 144

2 Proof. The upper bound is clear, as a binary tree of height h can only contain internal nodes. h 1 j=0 2 j = 2 c Ernst Mayr, Harald Räcke 145

3 Proof (cont.) Induction (base cases): 1. an AVL-tree of height h = 1 contains at least one internal node, 1 F 3 1 = 2 1 = an AVL tree of height h = 2 contains at least two internal nodes, 2 F 4 1 = 3 1 = 2 c Ernst Mayr, Harald Räcke 146

4 Induction step: An AVL-tree of height h 2 of minimal size has a root with sub-trees of height and h 2, respectiely. Both, sub-trees hae minmal node number. h 2 Let Then f h := 1 + minimal size of AVL-tree of height h. f 1 = 2 = F 3 f 2 = 3 = F 4 f = 1 + f h f h 2 1, hence f h = f h 1 + f h 2 = F h+2

5 Since F(k) 1 ( ) k 1 + 5, 5 2 an AVL-tree with n internal nodes has height Θ(log n). c Ernst Mayr, Harald Räcke 148

6 We need to maintain the balance condition through rotations. F this we ste in eery internal tree-node the balance of the node. Let denote a tree node with left child c l and right child c r. balance[] := height(t cl ) height(t cr ), where T cl and T cr, are the sub-trees rooted at c l and c r, respectiely. c Ernst Mayr, Harald Räcke 149

7 Rotations The properties will be maintained through rotations: z LeftRotate() z A RightRotate(z) C B C A B c Ernst Mayr, Harald Räcke 150

8 Double Rotations z y D A B C LeftRotate (y) RightRotate () z y y z D DoubleRightRotate () A B C A B C D

9 AVL-trees: Insert Insert like in a binary search tree. Let denote the parent of the newly inserted node. One of the following cases holds: a a bal() = 1 bal() = 0 bal() = 0 bal() = 1 If bal[] 0, T has changed height; the balance-constraint may be iolated at ancests of. Call fi-up(parent[]) to reste the balance-condition. c Ernst Mayr, Harald Räcke 152

10 AVL-trees: Insert Inariant at the beginning fi-up(): 1. The balance constraints holds at all descendants of. 2. A node has been inserted into T c, where c is either the right left child of. 3. T c has increased its height by one (otw. we would already hae abted the fi-up procedure). 4. The balance at the node c fulfills balance[c] { 1, 1}. This holds because if the balance of c is 0, then T c did not change its height, and the whole procedure will hae been abted in the preious step. c Ernst Mayr, Harald Räcke 153

11 AVL-trees: Insert Algithm 11 AVL-fi-up-insert() 1: if balance[] { 2, 2} then DoRotationInsert(); 2: if balance[] {0} return; 3: AVL-fi-up-insert(parent[]); We will show that the aboe procedure is crect, and that it will do at most one rotation. c Ernst Mayr, Harald Räcke 154

12 AVL-trees: Insert Algithm 12 DoRotationInsert() 1: if balance[] = 2 then 2: if balance[right[]] = 1 then 3: LeftRotate(); 4: else 5: DoubleLeftRotate(); 6: else 7: if balance[left[]] = 1 then 8: RightRotate(); 9: else 10: DoubleRightRotate(); c Ernst Mayr, Harald Räcke 155

13 AVL-trees: Insert It is clear that the inariant f the fi-up routine holds as long as no rotations hae been done. We hae to show that after doing one rotation all balance constraints are fulfilled. We show that after doing a rotation at : fulfills balance condition. All children of still fulfill the balance condition. The height of T is the same as befe the insert-operation took place. We only look at the case where the insert happened into the right sub-tree of. The other case is symmetric. c Ernst Mayr, Harald Räcke 156

14 AVL-trees: Insert We hae the following situation: h + 1 The right sub-tree of has increased its height which results in a balance of 2 at. Befe the insertion the height of T was h + 1. c Ernst Mayr, Harald Räcke 157

15 Case 1: balance[right[]] = 1 We do a left rotation at LeftRotate () h h Now, T has height h + 1 as befe the insertion. Hence, we do not need to continue. c Ernst Mayr, Harald Räcke 158

16 Case 2: balance[right[]] = 1 RightRotate () y y h 2 h 2 h 2 h 2 DoubleLeftRotate () y LeftRotate () Height is h + 1, as befe the insert. h 2 h 2

17 AVL-trees: Delete Delete like in a binary search tree. Let denote the parent of the node that has been spliced out. The balance-constraint may be iolated at, at ancests of, as a sub-tree of a child of has reduced its height. Initially, the node c the new root in the sub-tree that has changed is either a dummy leaf a node with two dummy leafs as children. c Case 1 Case 2 In both cases bal[c] = 0. Call fi-up() to reste the balance-condition. c Ernst Mayr, Harald Räcke 160

18 AVL-trees: Delete Inariant at the beginning fi-up(): 1. The balance constraints holds at all descendants of. 2. A node has been deleted from T c, where c is either the right left child of. 3. T c has either decreased its height by one it has stayed the same (note that this is clear right after the deletion but we hae to make sure that it also holds after the rotations done within T c in preious iterations). 4. The balance at the node c fulfills balance[c] = {0}. This holds because if the balance of c is in { 1, 1}, then T c did not change its height, and the whole procedure will hae been abted in the preious step. c Ernst Mayr, Harald Räcke 161

19 AVL-trees: Delete Algithm 13 AVL-fi-up-delete() 1: if balance[] { 2, 2} then DoRotationDelete(); 2: if balance[] { 1, 1} return; 3: AVL-fi-up-delete(parent[]); We will show that the aboe procedure is crect. Howeer, f the case of a delete there may be a logarithmic number of rotations. c Ernst Mayr, Harald Räcke 162

20 AVL-trees: Delete Algithm 14 DoRotationDelete() 1: if balance[] = 2 then 2: if balance[right[]] = 1 then 3: LeftRotate(); 4: else 5: DoubleLeftRotate(); 6: else 7: if balance[left[]] = {0, 1} then 8: RightRotate(); 9: else 10: DoubleRightRotate(); c Ernst Mayr, Harald Räcke 163

21 AVL-trees: Delete It is clear that the inariant f the fi-up routine holds as long as no rotations hae been done. We show that after doing a rotation at : fulfills balance condition. All children of still fulfill the balance condition. If now balance[] { 1, 1} we can stop as the height of T is the same as befe the deletion. We only look at the case where the deleted node was in the right sub-tree of. The other case is symmetric. c Ernst Mayr, Harald Räcke 164

22 AVL-trees: Delete We hae the following situation: h + 1 h The right sub-tree of has decreased its height which results in a balance of 2 at. Befe the insertion the height of T was h + 2. c Ernst Mayr, Harald Räcke 165

23 Case 1: balance[left[]] {0, 1} RightRotate () h h h h If the middle subtree has height h the whole tree has height h + 2 as befe the deletion. The iteration stops as the balance at the root is non-zero. If the middle subtree has height the whole tree has decreased its height from h + 2 to h + 1. We do continue the fi-up procedure as the balance at the root is zero.

24 Case 2: balance[left[]] = 1 LeftRotate () y y h 2 h 2 h 2 h 2 y RightRotate () Sub-tree has height h + 1, i.e., it has shrunk. The balance at y is zero. We continue the iteration. DoubleRightRotate () h 2 h 2