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CSE 326: Data Structures
AVL TreesBrian CurlessSpring 2008
Potential Balance ConditionsWhat are some candidate balance conditions on BST’s?1. Left and right subtreesof the root haveequal number of nodes2. Left and right subtreesof the root haveequal^ height
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Potential Balance Conditions3. Left and right subtreesof every node
have
equal number of nodes4. Left and right subtreesof^ every node have equal
height
Balancing Trees
-^ Many algorithms exist for keeping trees balanced– Adelson-Velskii and Landis (AVL) trees– Splay trees and other
self-adjusting
trees
– B-trees and other multiway search trees (for^ very^ large trees)• Today we will talk about
AVL trees
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The AVL Tree Data Structure
Ordering property–^ Same as for BSTStructural properties1. Binary tree property(0,1, or 2 children)2. Heights of left and rightsubtrees of
every node differ by at most 1 Result:Worst case depth of anynode is: O(log
n )^
Recursive Height Calculation Recall : height is max numberof edges from root to a leafWhat is the height at A?Note: height(null) = -
A
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AVL trees or not? 10
Proving Shallowness Bound
AA^0
1
A^2
A^3
A^4
Ais smallest AVL tree of height h. h^ Built from A
and A h -
attached to a new root. h -
(Note: these A
’s are not unique; e.g., could swap children.) h
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Testing the Balance Property
We need to be able to:1. Track Balance2. Detect Imbalance3. Restore Balance
What if we insert(30)?
An AVL Tree
0 (^01) 2
1 3
data 10 height 3 children
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AVL trees: find, insert
-^ AVL find
:
- same as BST find.• AVL insert
:
except^
may need to “fix”
the AVL tree after inserting new value.We will consider the 4 fundamental insertioncases…
Case #1: left-left insertion (zig-zig)
a
Z
b Y h^ X
h
h a
Z
b^ h Y
h
X
Insert on left child’s left h+
h+
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Case #1: repair with single rotation
a ZY b h+
h^
h
X < b < Y < a < Z
h+
a
Z
b^ h Y
h
h+2 X X
h+3 Single rotation
Height of tree before/after?
Effect on Ancestors?
Cost?^
Single rotation example^104
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Case #2: left-right insertion (zig-zag)
a
Z
b Y h+1 X
h
h a
Z
b h
h
X^
Insert on left child’s rightY h
h+
Case #2: repair with single rotation
a Z b h+ h
h
X < b < Y < a < Z^ Are we better off now?
a
Z
b h
h
X X
h+1 Y^ Single rotation Y h+
h+
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Double rotation, step 2
Case #3: right-left insertion (zig-zag)
a X
b
Z
c^ h-^
h
h
c
Z
a X
b h-^
h
h^
h h
V
U
V
h+1 U
h+ h+3 h+^
h+ h+
Double rotation
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Case #4: right-right insertion (zig-zig)
a X
b Y
h+ h h
Z
a X
b h Y
h+
h^
h+2 Z h+ h+
h+ Single rotation
AVL tree case summary Let^ x^ be the node where an imbalance occurs.Four cases to consider. The insertion below
a^ is in the
1.^ left^ child’s^ left
subtree_._
(zig-zig)
2.^ left^ child’s^ right
subtree_._ (zig-zag)
3.^ right
child’s^ left
subtree_._ (zig-zag)
4.^ right
child’s^ right
subtree_._ (zig-zig)
Cases 1 & 4 are solved by a single rotation:1.^ Rotate between
a^ and child Cases 2 & 3 are solved by a double rotation:1.^ Rotate between
a ’s child and grandchild
2.^ Rotate between
a^ and^ a ’s new child
29 (^95) 2
Single and Double Rotations:Inserting what integer values wouldcause the tree to need a:1. single rotation?2. double rotation?3. no rotation?
Insertion procedure
1.^ Find spot for new key2.^ Hang new node there with this key3.^ Search back up the path for imbalance4.^ If there is an imbalance:cases #1,#4: Perform single rotation and exitcases #2,#3: Perform double rotation and exitBoth rotations keep the subtree height unchanged.Hence only one rotation is sufficient!
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More insert examples
Insert(33)
Unbalanced?How to fix?
0
(^100) 1 0
2 3
0
Single Rotation^20 9 2
0
(^200) 1 0
3 4
1 0
0
(^310)
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More insert examples
0
(^110) 1 0
2 3
0 0 12
Insert(3) Unbalanced?How to fix?
Insert into an AVL tree: 5, 8, 9, 4, 2, 7, 3, 1
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AVL complexity
What is the worst case complexity of an insert?What is the worst case complexity of a find?
