AVL Trees

1
AVL Trees

• And the pain you must suffer to learn them

2
The AVL Property

• For every node the left and right side differ in
  height by less than two.
• The left and right side are themselves AVL trees.

3
How Balanced is That?

• AVL trees might end up kinda 'sparse'.
• The worst case height of an AVL tree with n nodes
  is about 1.44log(n).
• Discussion at http//ciips.ee.uwa.edu.au/morris/Y
  ear2/PLDS210/AVL.html

4
Rotations

• Rotations are to rearrange nodes to maintain
  AVLness.
• They are needed on insert and delete.

5
Names of Rotations

• There are four of them
• Single Left (LL) and Single Right (RR)
• Double Left (LR) and Double Right (RL)
• The names describe which way the node moves. For
  example, a left rotation moves the node down and
  left.
• Nodes always move down when rotated.

6
How To Do Single Rotations
7
A Left Rotation via Pointers

• tempp-gtright
• p-gtrighttemp-gtleft
• temp-gtleftp
• ptemp

8
Double Rotation

• A LR double rotation is to rotate something to
  the left, and then its former parent to the
  right.
• A RL double rotation is to rotate something to
  the right, and then its former parent to the left.

9
Rotation Examples

• All the rotations in picture form
  http//sky.fit.qut.edu.au/maire/avl/System/AVLTre
  e.html
• All the rotations in pictures and text
  http//www.cs.jcu.edu.au/Subjects/cp2001/1997/foil
  s/balancedTrees/balancedTrees.html
• Sample single and double rotationshttp//ironbark
  .bendigo.latrobe.edu.au/courses/subjects/DataStruc
  tures/mal/session200/lecture.html

10
The AVL Insertion Algorithm

• An Intuitive description can be found at
  http//www.ucfv.bc.ca/cis/watkissc/200009/200009Co
  mp175/notes/avlTrees.html

11
The AVL Game

• Play the game http//www.seanet.com/users/arsen/av
  ltree.html

12
The Insertion Algorithm

• Recursively call down to find the insertion
  point.
• Insert there (it will always be a leaf node).
• Call rebalance at the end.
• Will rebalance from the bottup up.
• Will rebalance only the path from the change to
  the root.

13
Insertion Code

• insert_node (inData)
• if (this NULL)
• return new NodeltTgt (inData)
• if (inData lt data)
• left left -gt insert_node (inData)
• else
• right right -gt insert_node
  (inData)
• return balance ()

14
Notes About the Code

• Calls balance on the way up.
• Only calls balance on the path between the change
  and the root.
• Call balance too many times for an insert.
• In this example ....We never change the parent
  node (or root).Instead we return what that
  parent node should be. Neat trick!

15
The Deletion Algorithm

• Find the place to delete.
• Swap with the in order successor.
• Delete that node
• Remember, he might have ONE kid.
• Rebalance on the way up from the node you just
  deleted.

16
Balancing

• Let d difference in height between kids.
• If (abs(d) lt 2) return
• If (d lt 0) // Heavy on the left
• if (right-gtdifference gt 0) // kid heavy on
  right
• right right-gtrotate_right()
• return rotate_left()
• .... // same stuff other side
• recompute my height
• return

17
Computing the Height

• Each node's height is just max(left-gtheight,
  right-gtheight) 1
• This only changes along the path from
  insertion/deletion to the root.
• The obvious recursive code searches the entire
  tree ... don't do that!