# lec4 - MIT OpenCourseWare http/ocw.mit.edu 6.006...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.006 Introduction to Algorithms Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008 Lecture 4: Balanced Binary Search Trees Lecture Overview • The importance of being balanced AVL trees • – Definition – Balance – Insert Other balanced trees • • Data structures in general Readings CLRS Chapter 13. 1 and 13. 2 (but different approach: red-black trees) Recall: Binary Search Trees (BSTs) • rooted binary tree each node has • – key – left pointer – right pointer – parent pointer See Fig. 1 65 41 20 11 50 29 26 3 2 1 1 φ φ φ Figure 1: Heights of nodes in a BST 1 Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008 x ≤x ≥x Figure 2: BST property • BST property (see Fig. 2). • height of node = length ( edges) of longest downward path to a leaf (see CLRS B.5 for details). The Importance of Being Balanced: • BSTs support insert, min, delete, rank, etc. in O ( h ) time, where h = height of tree (= height of root). • h is between lg( n ) and n : Fig. 3). vs. Perfectly Balanced Path Figure 3: Balancing BSTs balanced BST maintains h = O (lg n ) all operations run in O (lg n ) time. • ⇒ 2 Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008 AVL Trees: Definition...
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lec4 - MIT OpenCourseWare http/ocw.mit.edu 6.006...

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