AVL TREES Height Balance : AVL Trees h 1 h 2 | h - h | 1 AVL - - PDF document

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AVL TREES Height Balance : AVL Trees h 1 h 2 | h - h | 1 AVL - - PDF document

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AVL TREES Height Balance : AVL Trees h 1 h 2 | h - h | 1 AVL AVL 2 1 non-AVL trees AVL trees S. Prasitjutrakul 1994 AVL Trees : I nsertion m m k k k u u k k m m v t u m m t t u k m u k u p v t k p v v

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SLIDE 1

AVL TREES

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SLIDE 2
  • S. Prasitjutrakul 1994

Height Balance : AVL Trees

AVL AVL

h2 h1 | h - h | ≤ 1

1 2

AVL trees non-AVL trees

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SLIDE 3
  • S. Prasitjutrakul 1994

AVL Trees : I nsertion

k k m m k u k m u m k u t v m k u t v p m k t p u v t m k p u v

Rotations

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SLIDE 4
  • S. Prasitjutrakul 1994

Rotations : Right High (Left Rotation)

A B C r x h h h+1 Total height = h+3 C B A r x h h h+1 Total height = h+2

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SLIDE 5
  • S. Prasitjutrakul 1994

Rotations : Left High (Double Rotation)

Total height = h+3

B C r x A h D h w h h C w x D h A r B h h h

Total height = h+2

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SLIDE 6
  • S. Prasitjutrakul 1994

Rotations : Left High (Double Rotation)

Total height = h+3

B C r x A h D h w h h-1 C w x D h A r B h h h-1

Total height = h+2

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SLIDE 7
  • S. Prasitjutrakul 1994

Rotations : Left High (Double Rotation)

Total height = h+3

B C r x A h D h w h-1 h C w x D h A r B h-1 h h

Total height = h+2

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SLIDE 8
  • S. Prasitjutrakul 1994

Deletion from an AVL Tree

A B A B A B A B B C A B C A

p p p p p q p q

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SLIDE 9
  • S. Prasitjutrakul 1994

Deletion from an AVL Tree

B C A B C A C D B A B A D C

p q p q p q r r p q

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SLIDE 10
  • S. Prasitjutrakul 1994

Deletion : Example

a g f e d c b k j h l i u p t s r

  • n

m

deleted

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SLIDE 11
  • S. Prasitjutrakul 1994

Deletion : Example

a g f e d c b k j h l i t s r u

  • n

m

deleted

p s r

  • n
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SLIDE 12
  • S. Prasitjutrakul 1994

Deletion : Example

t s r u

  • n

m

deleted

A

t s r u

  • n

m

shorter

A

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SLIDE 13
  • S. Prasitjutrakul 1994

Deletion : Example

t s r u

  • n

m

shorter

A

t s r u

  • n

m

shorter

A

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SLIDE 14
  • S. Prasitjutrakul 1994

Deletion : Example

t s r u

  • n

a g f e d c b k j h l i m

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SLIDE 15
  • S. Prasitjutrakul 1994

Deletion : Example

t s r u

  • n

a e d c b j k l g f h i m

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SLIDE 16
  • S. Prasitjutrakul 1994

The Height of an AVL Tree

กํ าหนดให F คือ AVL tree ที่มีความสูง h ที่มีจํ านวน nodes นอยที่สุด และให | F | คือจํ านวน nodes ของตนไม F

h h h

F

h

F h

F h F h

Fibonacci Tree

worst-case bound