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cs2851-Balanced Trees

Course: CS 2851, Fall 2009
School: Milwaukee School of...
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Binary Balanced Search Trees CS-2851 Dr. Mark L. Hornick 1 For any balanced binary search tree, searching, inserting and deleting have worstTime(n) that is logarithmic in n But insertions and deletions can result in a tree that is highly unbalanced 100 100 O(log n) performance drifts toward O(n) 40 80 160 80 90 120 200 40 90 A BST is balanced only if if its height is logarithmic in n 50 50 N=8,...

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Binary Balanced Search Trees CS-2851 Dr. Mark L. Hornick 1 For any balanced binary search tree, searching, inserting and deleting have worstTime(n) that is logarithmic in n But insertions and deletions can result in a tree that is highly unbalanced 100 100 O(log n) performance drifts toward O(n) 40 80 160 80 90 120 200 40 90 A BST is balanced only if if its height is logarithmic in n 50 50 N=8, h=log2(8) = 3 CS-2851 Dr. Mark L. Hornick N=5, h > log2(5) = 1.6 2 An AVL tree is a balanced BST that either is empty or in which eights between left and right subtrees differs by only 1 Unbalanced vs. balanced: 160 100 100 200 80 160 80 120 40 90 120 200 40 90 50 50 he left and right subtrees are also AVL trees. CS-2851 Dr. Mark L. Hornick 3 Another (balanced) AVL Tree 20 10 30 5 32 35 CS-2851 Dr. Mark L. Hornick 4 More AVL Tree examples 50 70 10 20 50 70 10 20 50 70 90 25 61 40 CS-2851 Dr. Mark L. Hornick 5 Non-AVL Tree examples 50 70 61 20 10 25 40 CS-2851 Dr. Mark L. Hornick 6 50 70 10 20 50 70 80 90 Balance can be maintained through rotations Rotation: an adjustment to the tree, around an element, that maintains the required ordering of elements Right rotation around element 90 90 45 45 20 90 20 CS-2851 Dr. Mark L. Hornick 7 For any right rotation around element x, the right subtree of xs left child becomes x y the left subtree of x. y x z z Rotate right around 100: 100 ? 80 120 40 50 90 CS-2851 Dr. Mark L. Hornick 8 For any right rotation around element x, the right subtree of xs left child becomes x y the left subtree of x. y x z z Here is a right rotation around 100: 100 80 80 120 40 100 40 90 50 90 120 50 Notice that 90 is now in the right subtree. CS-2851 Dr. Mark L. Hornick 9 Rotate about right 20 20 10 35 5 12 7 CS-2851 Dr. Mark L. Hornick 10 Right rotation about 20 20 10 35 5 12 35 10 20 5 12 7 CS-2851 Dr. Mark L. Hornick 7 11 For a right rotation around x x.left = lc.right; lc.right = x; where x is a reference to an Entry object, and lc is a reference to the left child of x Here is a right rotation around 100: 100 80 80 120 40 100 40 90 50 90 120 50 Notice that 90 is now in the right subtree. CS-2851 Dr. Mark L. Hornick 12 In a rotation around x, the only restructuring is to the subtree rooted at x right rotation around 100: 160 160 100 200 80 200 80 120 40 100 40 50 90 CS-2851 Dr. Mark L. Hornick 50 90 120 13 But the parent of x has to be adjusted (text p 421): private void rotateRight(Entry<E> x) { Entry<E> lc = x.left; x.left = lc.right; // From previous slide if (lc.right != null) lc.right.parent = x; lc.parent = x.parent; if (x.parent == null) // x is root root = lc; else if (x.parent.right == x) // x is right child x.parent.right = lc; else // x is left child x.p...

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