1. General Trees 2. Binary Search Trees 3. AVL Trees 4. Heap Trees.
AVL Trees Balanced Binary Search Trees (not covered in book, but related to pp 165-169)
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AVL Trees Balanced Binary Search Trees (not covered in book, but related to pp 165-169)
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Transcript of AVL Trees Balanced Binary Search Trees (not covered in book, but related to pp 165-169)
AVL Trees
Balanced Binary Search Trees(not covered in book, but related to
pp 165-169)
AVL Trees
• First-invented self-balancing binary search tree
• Named after its two inventors, 1. G.M. Adelson-Velsky and
2. E.M. Landis, – published it in their 1962 paper "An algorithm
for the organization of information."
AVL Properties
• First, its a binary search tree...
• L <= P and P <= R
P
L R
Question?
• Is this a binary tree search tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
BTW, Binary Heap Properties
• Min Heap• P <= L and P <= R
• Max Heap• P >= L and P >= R
P
L R
AVL Properties
• An AVL tree is a balanced binary tree
• To understand balance we need to understand the notion of Tree Height
55
32 71
64 86 Height 0
Height 1
Height 2
AVL Properties
• By default, nodes with no children have a height of Height of 0.
55
32 71
64 86 Height 0
Height 1
Height 2
Height 0
Height 0
AVL Properties
• But, we must also understand the concept of Sub-trees
55
32 71
64 86
Height 0 Height 1
Height 2
Height 0 Height 0
sub-tree L has a height of 0 sub-tree R has a
height of 1
Height = max(L.height, R.height) + 1
AVL Properties
• Also empty sub-trees have a Height of -1
44
58
91
Height = 2 = max(0, 1) + 1
Height = 0 = max(-1,-1) + 1
Height = max(L.height, R.height) + 1
Height = 1 = max(-1, 0) + 1
AVL Properties
• Anyway, the AVL Balance Property is as follows...
• For ALL nodes, the Height of the Left and Right Sub-trees can only differ by 1.
P A Node
L R
1.. heightRheightL
AVL Properties
• Wouldn’t this be a better Balance property?
• For ALL nodes, the Height of the Left and Right Sub-trees must be equal!
P A Node
L R
0.. heightRheightL
AVL Properties
• For ALL nodes, the Height of the Left and Right Sub-trees must be equal!
1
3
715
31
AVL Properties
• For ALL nodes, the Height of the Left and Right Sub-trees must be equal!
• Strict Balance Conditions are too restrictive. 1
3
715
31
Question?
• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
Question?
• Is this an AVL Tree?
3 9
5
15 21
17
30 34
32
43 49
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
Question?
• Is this an AVL Tree?
3 9
5
15 21
17
30 34
32
43 49
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
1 1 1 1 1 1
122 2
3
4
3
Question?
• Is this an AVL Tree?
3
5 17 32 79
7712 35 56
29 70
50
Question?
• Is this an AVL Tree?
3
5 17 32 79
7712 35 56
29 70
50
1 0 0 0
102 1
3
4
2
0
Question?
• Is this an AVL Tree?
3
5 17 32 79
7712 35 56
29 70
50
1 0 0 0
102 1
3
4
2
00
+1
+1 +1
+1
+1
-1
-10
00 0
Question?
• Is this an AVL Tree?
3 9
5
15 21
17
49
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
Question?
• No
3 9
5
15 21
17
49
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
Question?
• Did this fix the problem?
3 9
5
15 21
17 49
46
52 54
53
58 68
60 71 79
7712
35
56
29 70
50
Question?• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
69
Question?• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
690
-1
-1
-1
+2
Question?• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60 71 79
7712 35 56
29 70
50
690
-1
-1
-1
+2
Question?• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53
58 68
60
71 79
7712 35 56
29
70
50
690
-1
-1
-1
+2
Question?• Is this an AVL Tree?
3 9
5
15 21
17
28 34
32
43 51
46
52 54
53 58 68
60
71 79
77
12 35 56
29
70
50
69
0
Correcting Imbalance
1. After every insertion
2. Check to see if an imbalance was created.• All you have to do backtrack up the tree
3. If you find an imbalance, correct it.
4. As long as the original tree is an AVL tree, there are only 4 types of imbalances that can occur.
Imbalances
• Left-Left (Single Rotation)
• Right-Right(Single Rotation)
• Left-Right (Double Rotation)
• Right-Left(Double Rotation)
Illustration
• http://www.eli.sdsu.edu/courses/fall96/cs660/notes/avl/avl.html
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