The document discusses binary search trees and their variants. It explains that search trees are important for algorithm design and it is desirable to minimize the search time of each node. There are static and dynamic binary search trees, with the latter adjusting its structure during access. AVL trees are a type of self-balancing binary search tree where rotations are used to rebalance the tree after insertions or deletions and ensure the heights of subtrees differ by at most one. Compilers use symbol tables implemented as search trees to track variables in source code.

1. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
1

2. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
2
 Search trees are of great importance in an
algorithm design
 It is always desirable to keep the search time of
each node in the tree minimal

3. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
3
 Variations in binary search trees: static and dynamic
 Ways of building trees of each type to guarantee that
the trees remain balanced

4. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
4
 BSTs are widely used for retrieving data from
databases, look-up tables, and storage dictionaries
 It is the most efficient search technique having time
complexity that is logarithmic in the size of the set

5. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
5
 These two cases lead to the following two kinds of
search trees:
 Static BST
 Dynamic BST

6. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
6
 Static BST is the one that is not allowed to update
its structure once it is constructed
 In other words, the static BST is an offline
algorithm, which is presumably aware of the access
sequence beforehand
Static BST

7. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
7
 A dynamic BST is the one that changes during the
access sequence
 We assume that the dynamic BST is an online
algorithm, which does not have prior information
about the sequence
Dynamic BST

8. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
8
 While compilers and assemblers are scanning a
program, each identifier must be examined to
determine if it is a keyword
 This information concerning the keywords in a
programming language is stored in a symbol table
 Symbol table is a kind of ‘keyed table’
 The keyed table stores <key, information> pairs with
no additional logical structure

9. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
9
 The operations performed on symbol tables are the
following:
 Insert the pairs <key, information> into the
collection
 Remove the pairs <key, information> by specifying
the key
 Search for a particular key
 Retrieve the information associated with a key

10. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
10
 There are two different techniques for implementing
a keyed table: symbol table and tree table
 Static Tree Tables
 Dynamic Tree Tables

11. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
11
 Static Tree Tables
 When symbols are known in advance and no
insertion and deletion is allowed, it is called a static
tree table
 An example of this type of table is a reserved word
table in a compiler

12. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
12
 To optimize a table knowing what keys are in the table and
what the probable distribution is of those that are not in the
table, we build an optimal binary search tree (OBST)
 Optimal binary search tree is a binary search tree having an
average search time of all keys optimal
 An OBST is a BST with the minimum cost

13. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
13
 A dynamic tree table is used when symbols are not
known in advance but are inserted as they come and
deleted if not required
 Dynamic keyed tables are those that are built on-the-
fly
 The keys have no history associated with their use

14. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
14
 An AVL tree is a BST where the heights of the left and
right subtrees of the root differ by utmost 1 and the
left and right subtrees are again AVL trees
 The formal definition is as follows:
 An empty tree is height-balanced if T is a non-
empty binary tree with TL and TR as its left and right
subtrees, respectively

15. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
15
 The balance factor of a node T, BF(T), in a binary tree is
hL − hR, where hL and hR are the heights of the left
and right subtrees of T, respectively
 For any node T in an AVL tree,
 the BF(T) is equal to −1, 0, or 1

16. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
16
 For example, consider the BST as shown in Fig
BF(Fri) = 0
BF(Mon) = +1
BF(Sun) = +2
 Because BF(Sun) = +2, the tree is no longer height-
balanced, and it should be restructured
Unbalanced BST

17. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
17
 In an AVL tree, after insertion of each node, it is
checked whether the tree is balanced or not
 If unbalanced, it is rebalanced immediately
 A node is inserted or deleted from a balanced tree,
then it may become unbalanced

18. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
18
 So to rebalance it, the position of some nodes can
be changed in proper sequence
 This can be achieved by performing rotations of
nodes
 Rebalancing of AVL tree is performed using one of
the four rotations:
 LL,RR,LR,RL

19. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
19
Balancing a tree by rotating towards right (a) Unbalanced Balanced tree

20. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
20
Balancing a tree by rotating towards left (a) Unbalanced tree (b)
Balanced tree
Example

21. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
21
Repetition Construct
Case 1: LL (Left of Left)
Consider the BST in Fig :

22. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
22
Case 2: RR (Right of Right)

23. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
23
Case 3: RL (Right of Left)
Case LR for unbalanced tree due to insertion in right of left of a node
(a)

24. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
24
 Case 3: RL (Left to right)
Case LR for unbalanced tree due to insertion in right of left of a node (a)
Scenario 1 (b)

25. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
25
Case 3: RL (Right of Left)
Case LR for unbalanced tree due to insertion in right of left of a node
(a) Scenario 1 (b) Scenario 2 (c) Scenario 3

26. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
26
 Case 4: LR (Left of Right)
Case RL for unbalancing due to insertion in left of right of a node (a)

27. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
27
Case RL for unbalancing due to insertion in left of right of a node (a)
Scenario 1 (b)
 Case 4: LR (Left of Right)

28. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
28
Case RL for unbalancing due to insertion in left of right of a node (a)
Scenario 1
(b) Scenario 2 (c) Scenario 3
Case 4: LR (Left of Right)

29. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
29
 Insertions and deletions in AVL tree are performed as
in BSTs and followed by rotations to correct the
imbalances in the outcome trees
 Unbalancing of an AVL tree due to insertion is removed
in a single rotation
 However, imbalancing due to the deletion may require
multiple steps for balancing

30. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
30
 Figure demonstrates the deletion of a node in a given AVL
tree
(a) Original tree

31. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
31
(b) Delete 4

32. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
32
(c)Note the imbalance at node 3 implies an LL rotation around node 2

33. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
33
(d) Imbalance at node 5 implies a RR rotation around node 8

34. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
34
 Search trees are of great importance in an algorithm design.
 It is always desirable to keep the search time of each node in the tree
minimal.
 OBST maintains the average search time of all the nodes optimal.
 In an AVL tree, after insertion of each node, it is checked whether the tree is
balanced or not.
 If unbalanced, it is rebalanced immediately.
 Rebalancing of AVL tree is performed using one of the four rotations: LL, RR,
LR, RL.
 AVL trees work by insisting that all nodes of the left and right subtrees differ
in height by utmost 1, which ensures that a tree cannot get too deep.
 Compilers use hash tables to keep track of the declared variables in a source
code called as a symbol table.
 Imbalancing of an AVL tree due to insertion is removed in a single rotation.
However, Imbalancing due to the deletion may require multiple steps for
balancing.

35. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
35