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Lecture13: Tree III Bohyung Han CSE, POSTECH bhhan@postech.ac.kr CSED233: Data Structures (2014F)

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion Order matters Difference between these binary search trees? Insert: 4, 2, 1, 3, 6, 5, 7 Insert: 1, 2, 3, 4, 5, 6, 7 2 4 2 31 6 75 1 5 3 2 4 7 6

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Complexity of Operations 3

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Self‐balancing Trees Variants of binary search tree Self‐balancing Additional rules for how to structure tree Needs operations to move nodes around Constraints on height differences of leaves Ensures that tree does not degenerate to list. 4

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 AVL Tree 5 44 17 32 48 78 8850 62 4 2 1 3 1 2 11 AVL tree

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion Insertion is as in a binary search tree. 6 44 17 32 48 78 8850 62 4 2 1 3 1 2 11 Before insertion After insertion 44 17 32 48 78 8850 62 5 2 1 4 1 3 21 54 1

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion 7 44 17 32 48 78 8850 62 5 2 1 4 1 3 21 54 1 b=x a=y c=z

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Restructuring Single Rotation: 8 h h-1 h-2 h-3 h-1 h-2 h-3 h-2

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Restructuring Double rotations: 9 h h-1 h-2 h-3 h-2 h-1 h-3 ≤h-3 h-2 ≤h-3

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion Example 10 44 17 32 48 78 8850 62 5 2 1 4 1 3 21 54 1 b=x a=y c=z After insertion unbalanced After the first rotation 44 17 32 50 78 8862 54 5 2 1 4 1 3 1 2 48 1 a=y b=x c=z

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion Example 11 44 17 32 48 62 88 50 54 4 2 1 3 2 2 11 78 1 c=z a=y b=x After the first rotation After the second rotation 44 17 32 50 78 8862 54 5 2 1 4 1 3 1 2 48 1 a=y b=x c=z

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Removal 12 44 17 32 48 62 88 50 54 4 2 1 3 2 2 11 78 1 Before deletion of 32 44 17 48 62 88 50 54 4 1 3 2 2 11 78 1 After deletion of 32

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Reconstructing after a Removal 13 44 17 48 62 88 50 54 4 1 3 2 2 11 78 1 z x y w

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Reconstructing after a Removal Recursive reconstructing Restructuring may upset the balance of another node higher in the tree. We must continue checking for balance until the root is reached. 14 44 17 48 62 88 50 54 4 1 3 2 2 11 78 1 z w y w 62 44 48 78 50 54 4 3 2 1 2 11 88 1 y x z 17 x

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CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 AVL Tree Performance 15

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