AVL Balanced Trees

1
AVL Balanced Trees

• Introduction
• Data structure
• AVL tree balance condition
• AVL node level
• AVL rotations
• Choosing the rotation
• Performing a balanced insertion
• Outputting data
• Testing and results

2
Introduction 1

• There is no reason why a Knuth binary tree should
  balance. Average performance for random input
  should not be greatly below balanced performance,
  but if the tree is made from sorted input, it
  will degrade to a linked list with access time of
  O(N) where there are N nodes. Access time for a
  node within a balanced tree will be log2N or
  slightly less.
• Operations which change the balance of a tree are
  insertion and deletion. Insertion might increase
  the maximum depth of a tree, but deletion can't
  do this.
• Therefore, if effort is going to be expended in
  balancing an unbalanced tree, by rearranging
  nodes, this can initially be most usefully
  applied following insertion of nodes.

3
Introduction 2

• A perfectly balanced tree can only exist if N is
  2x1 where x is an integer, i.e. if N is a value
  the series 1, 3, 7, 15, 31, 63, 127, 255 ... .
  The need for perfect balance could be relaxed by
  allowing the bottom level of the tree to be
  partially filled. This is difficult to achieve,
  as the entire tree might need rearranging after
  every insertion and deletion.
• The imbalance problem only arises within the path
  through the tree on which a node is inserted. A
  recursive insertion function will return upwards
  through the insertion path. If information about
  the maximum depths of nodes is maintained as part
  of each nodes structure, it is possible to
  correct imbalances through localised
  rearrangements following insertion on the
  recursive return path upwards to the root.

4
Data Structure
5
AVL Tree Balance Condition

• A particular approach to performing the tree
  rearrangements needed to keep a tree balanced was
  designed by AdelsonVelskii and Landis and named
  as the AVL tree.
• An AVL tree has a balance condition such that
  each node stores the maximum depth of nodes below
  it. A NULL tree has a level of 1 and a leaf node
  has a level of 0. A node having 1 or 2 subtrees
  will have as its own level the maximum of left or
  right subtree levels 1. Within an AVL tree, the
  maximum difference allowed between left and right
  subtree levels is 1. When an insertion or
  deletion operation results in this difference
  increasing to 2, rearrangements called rotations
  are performed to reduce the imbalance for an AVL
  tree to retain the AVL balance condition.

6
AVL Node Level

• Following an insertion or deletion which may
  influence the levels of
• subtrees, the level of nodes affected will need
  to be recalculated. This can be achieved using a
  function operating on the above data structure as
  follows

7
AVL Rotations

• A difference of levels between left and right
  subtrees can be increased, through insertion of a
  new node into the tree, in one of 4 possible
  ways
• 1. Insertion into left subtree of left
  branch.
• 2. Insertion into right subtree of left
  branch.
• 3. Insertion into left subtree of right
  branch.
• 4. Insertion into right subtree of right
  branch.
• The function prototype for a rotation will be of
  theform
• void rotN(TOP t) / N is rotation type 1-4 /

8
Rotation case 1
9
Rotation case 1
10
Rotation case 2
11
Rotation case 2
12
Rotation case 3
13
Rotation case 3
14
Rotation case 4
15
Rotation case 4
16
Choosing the rotation slide 1
The rotation type 1 4 can be determined by
comparing 3 keys, the key inserted and the child
and parent key. It is useful to have a 2 way
comparison function similar to the strcmp()
function in string.h.
17
Choosing the rotation slide 2
18
Calling the rotation
19
Performing an AVL balanced insertion 1 of 3
20
Performing an AVL balanced insertion 2 of 3
21
Performing an AVL balanced insertion 3 of 3
22
Output of AVL organised data 1

• Printing the entire tree can be done recursively
  in search order. It is useful when debugging to
  be able to identify the immediate left and right
  child nodes for each record printed.
• define DEBUG 1 / or 0 for less noisy output /
• Using a printdata function enables the printall
  function to be data independent.
• void printdata(DATA d) / prints DATA item d /
• printf("c ",d)

23
Output of AVL organised data 2
24
Testing code
25
Test Results

• AVL imbalance -2 AVL type 4
• AVL imbalance -2 AVL type 3
• AVL imbalance 2 AVL type 2 ...
• o insert duplicate key error
• ... (run the program avltree.c to see full o/p)?
• a 0 a b 2 c c 1 d d 0
• b e 3 g f 0 f g 1 h h 0
• e i 4 k j 0 j k 2 m l 0
• l m 1 i n 5 u o 1 p p 0
• o q 2 s r 0 r s 1 t t 0
• q u 3 w v 0 v w 2 y x 0
• x y 1 z z 0
• These results show in 4 columns per node
• i. The left child key or blank. ii. The node key.
  
• iii. The node level. iv. The right key or
  blank.

26
Recommended Reading

• "Data Structures and Algorithms in C",
• Mark Allen Weiss, Addison Wesley Chapter 4.