Balanced Trees

1
Balanced Trees

• There are several ways to define balance
• Examples
• Force the subtrees of each node to have almost
  equal heights
• Place upper and lower bounds on the heights of
  the subtrees of each node.
• Force the subtrees of each node to have similar
  sizes (number of nodes)

2
AVL Trees

• AVL tree
• A binary search tree
• with the property
• for every node, the heights of the left and right
  subtrees differ at most by one.
• Implementation issues
• Each node contains a value (-1, 1, 0) indicating
  which subtree is "heavier"
• Insert and Delete are modified. They restructure
  the tree to make it balanced (if necessary).

3
AVL trees
1
nodes are marked with balance value
1
-1
0
1
0
0
4
AVL trees Fixing imbalances

• An imbalance is detected when the height
  difference between two subtrees of a node becomes
  greater than 1 or smaller than -1.
• There are two types of imbalances

or
or
TYPE 1
TYPE 2
5
AVL trees Fixing imbalances

• Fixing an imbalance is done by rotating the tree.
• There are two types of rotation
• single rotation
• for TYPE 1 imbalances
• double rotation
• for TYPE 2 imbalances
• consists of two single rotations.
• The rotations must always preserve the BST
  property.

6
AVL Trees single rotation
node with imbalance
6
D
4
4
right rotate at node 6
C
6
2
2
A
B
C
D
B
A
This is a single right rotation. A single left
rotation is symmetric.
7
AVL Trees Double rotation
node with imbalance
node with imbalance
6
6
4
D
D
4
2
6
2
STEP2 right rotate at node 6
C
A
A
B
C
D
2
4
B
A
C
B
STEP 1 left rotate at node 2
8
AVL Trees Double rotation
If you want to do it in one step, imagine taking
4 and moving it up in between 2 and 6, so that 2
and 6 become its new children
node with imbalance
6
4
D
2
6
2
A
4
A
B
C
D
C
B
B and C will then be adopted by 2 and 6
respectively, in order to maintain the BST
property.
9
AVL Trees Insert

• Insert the node as in a BST
• Starting at the newly inserted node, travel up
  the tree (towards the root), updating the
  balances along the way.
• If the tree becomes unbalanced, decide which
  rotation needs to be performed, rotate the tree
  and update the balances.

10
AVL Trees Delete

• Delete the node as in a BST
• Starting at the newly inserted node, travel up
  the tree (towards the root), updating the
  balances along the way.
• If the tree becomes unbalanced, decide which
  rotation needs to be performed, rotate the tree
  and update the balances.
• Keep traveling towards the root, checking the
  balances. You may need to rotate again.

11
AVL Trees Insert/Delete

• Insert
• maximum possible number of rotations 1
• Delete
• maximum possible number of rotations lgn
• Worst case times
• search O(lgn)
• insert O(lgn)
• delete O(lgn)
• one rotation O(1)

12
AVL Trees Efficiency

• It can be shown that the worst case height of an
  AVL tree is at most 44 larger than the minimum
  possible for a BST (i.e. approximately 1.44lgn)

13
Other balanced trees

• Red-black trees
• Each node has an extra bit to hold a color
• The tree stays balanced by placing restrictions
  on the way the nodes are colored
• Every path from the root to a leaf has the same
  number of black nodes and there cannot be
  consecutive red nodes.
• A red-black tree is balanced when the longest
  path from the root to a leaf is at most twice the
  length of the shortest path from the root to a
  leaf.
• A red-black tree with n nodes has height at most
  2lg(n1)

14
Other balanced trees

• a-balanced trees
• a is a constant between 1/2 and 1
• A tree is a-balanced if, for every node
  xsizeleft(x) ? asizexsizeright(x) ?
  asizex