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CSED233: Data Structures (2014F)
Lecture13: Tree III
Bohyung Han
CSE, POSTECH
[email protected]
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
1
2
4
2
1
3
6
3
5
4
7
5
6
7
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Complexity of Operations
• Lookup Time
 Balanced tree: 𝑂(log 𝑛)
 Unbalanced tree: 𝑂(𝑛)
• Insertion time
 Balanced tree: 𝑂(log 𝑛)
 Unbalanced tree: 𝑂(𝑛)
• Balance does matter for efficient trees.
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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.
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
AVL Tree
• What is AVL tree?
 A self‐balancing binary search tree
 Adelson‐Velskii and Landis' tree,
named after the inventors
 For every node 𝑣 of tree 𝑇, the
heights of the children of 𝑣 differ
by at most 1.
 Height: 𝑂(log 𝑛)
44
2
4
78
17
1
3
2
32
1
50
1
48
88
62
1
AVL tree
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Insertion
• Insertion is as in a binary search tree.
44
2
44
4
78
17
1
3
2
32
48
1
50
1
1
4
3
32
88
62
78
17
1
1
50
1
2
5
88
48
62
2
1
54
Before insertion
After insertion
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Insertion
• Tree height after insertion
 All nodes along the path increase
their height by 1.
 It may violate the AVL tree
property.
• We rebalance the subtree of 𝑧
using a restructuring method.
 𝑧: the first violation node from
the bottom along the path
 𝑦: 𝑧’s child with the higher height
 𝑥: 𝑦’s child with the higher height
 𝑎, 𝑏, 𝑐 : inorder traversal of
node 𝑥, 𝑦, and 𝑧
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44
2
5
78
17
1
4
c=z
3
32
1
50
88
a=y
1
48
62
2
b=x
1
54
CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Restructuring
• Single Rotation:
h-1
h
h-3
a=z
h-2
b=y
h-3
T0
h-2
h-1
single rotation
h-3
c=x
T1
T3
T2
c=z
b=y
a=z
c=x
h-3
T0
single rotation
b=y
h-2
T1
T3
T2
b=y
a=x
c=z
a=x
T0
T1
T2
T3
T3
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T2
T1
T0
CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Restructuring
• Double rotations:
h
h-3
h-1
h-1
a=z
double rotation
h-2
c = y h-3
≤h-3 b = x ≤h-3
T0
T1
h-2
b=x
a=z
h-3
T3
T2
h-2
T0
a=y
T2
T1
double rotation
c=z
c = y h-3
≤h-3
T3
b=x
a=y
c=z
b=x
T3
T2
T0
T3
T2
T1
T0
T1
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Insertion Example
44
2
5
44
78
17
1
4
c=z
3
32
2
78
17
1
1
50
48
62
c=z
1
62
88
b=x
a=y
1
4
3
32
88
5
2
2
b=x
50
a=y
1
1
48
54
54
1
After the first rotation
After insertion unbalanced
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Insertion Example
44
2
5
78
17
1
4
c=z
3
32
44
2
1
62
88
62
17
1
b=x
2
48
54
b=x
2
50
78
a=y
a=y
1
3
2
32
50
4
1
1
48
54
c=z
1
1
88
After the second rotation
After the first rotation
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Removal
44
2
4
44
62
17
1
3
2
32
48
62
17
2
50
78
54
3
2
2
50
1
1
4
1
1
1
88
Before deletion of 32
48
78
54
1
1
88
After deletion of 32
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
Reconstructing after a Removal
• Node definition
 𝑧: the first unbalanced node encountered while travelling up the tree
from the violating node 𝑤
 𝑦: the child of 𝑧 with the larger height
 𝑥: the child of 𝑦 with the larger height
4
44
1
17
z
62
w
3
y
2
2
50
1
48
13
78
54
x
1
1
88
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.
44
1
17
4
z
62
62
w
3
2
2
50
1
48
3
y
78
54
1
x
1
1
17
44
88
14
y
78
z
w
1
4
50
48
2
x
1
2
88
54
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
AVL Tree Performance
• Restructuring
 A single restructure takes 𝑂(1) time.
 Using a linked‐structure binary tree
• Search
 A search takes 𝑂(log 𝑛) time.
 The height of tree is 𝑂(log 𝑛), no restructuring is needed.
• Insertion
 An insertion takes 𝑂(log 𝑛) time.
 Restructuring up the tree, maintaining heights is 𝑂(log 𝑛).
• Deletion
 A deletion takes 𝑂(log 𝑛) time.
 Restructuring up the tree, maintaining heights is 𝑂(log 𝑛).
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2014
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