The document discusses operations and applications of binary trees. Some key points: - There are several common operations defined on binary trees like accessing the left/right children or parent of a node. - Binary trees can be used to efficiently search for duplicates in a list by building the tree in a way that compares each new number to the root. - Traversing a binary tree can be done in preorder, inorder, or postorder by following specific steps for visiting each node.

1. Lecture # 10Lecture # 10

2. Operations on Binary TreeOperations on Binary Tree
 There are a number of operations that canThere are a number of operations that can
be defined for a binary tree.be defined for a binary tree.
 IfIf pp is pointing to a node in an existing tree thenis pointing to a node in an existing tree then
 left(left(pp)) returns pointer to the left subtreereturns pointer to the left subtree
 right(right(pp)) returns pointer to right subtreereturns pointer to right subtree
 parent(parent(pp)) returns the father ofreturns the father of pp
 brother(brother(pp)) returns brother ofreturns brother of pp..
 info(p)info(p) returns content of the node.returns content of the node.

3. Operations on Binary TreeOperations on Binary Tree
 In order to construct a binary tree, theIn order to construct a binary tree, the
following can be useful:following can be useful:
 setLeft(p,x)setLeft(p,x) creates the left child node of p.creates the left child node of p.
The child node contains the info ‘x’.The child node contains the info ‘x’.
 setRight(p,x)setRight(p,x) creates the right child node ofcreates the right child node of
p. The child node contains the info ‘x’.p. The child node contains the info ‘x’.

4. Applications of Binary TreesApplications of Binary Trees
 A binary tree is a useful data structure when two-A binary tree is a useful data structure when two-
way decisions must be made at each point in away decisions must be made at each point in a
process.process.
 For example, suppose we wanted to find allFor example, suppose we wanted to find all
duplicates in a list of numbers:duplicates in a list of numbers:
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 514, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

5. Applications of Binary TreesApplications of Binary Trees
 One way of finding duplicates is to compareOne way of finding duplicates is to compare
each number with all those thateach number with all those that precedeprecede it.it.
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 514, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 514, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

6. Searching for DuplicatesSearching for Duplicates
 If the list of numbers is large and is growing, thisIf the list of numbers is large and is growing, this
procedure involves a large number ofprocedure involves a large number of
comparisons.comparisons.
 A linked list could handle the growth but theA linked list could handle the growth but the
comparisons would still be large.comparisons would still be large.
 The number of comparisons can be drasticallyThe number of comparisons can be drastically
reduced by using a binary tree.reduced by using a binary tree.
 The tree grows dynamically like the linked list.The tree grows dynamically like the linked list.

7. Searching for DuplicatesSearching for Duplicates
 The binary tree is built in a special way.The binary tree is built in a special way.
 The first number in the list is placed in a nodeThe first number in the list is placed in a node
that is designated as the root of a binary tree.that is designated as the root of a binary tree.
 Initially, both left and right subtrees of the root areInitially, both left and right subtrees of the root are
empty.empty.
 We take the next number and compare it with theWe take the next number and compare it with the
number placed in the root.number placed in the root.
 If it is the same then we have a duplicate.If it is the same then we have a duplicate.

8. Searching for DuplicatesSearching for Duplicates
 Otherwise, we create a new tree node and put theOtherwise, we create a new tree node and put the
new number in it.new number in it.
 The new node is made the left child of the rootThe new node is made the left child of the root
node if the second number is less than the one innode if the second number is less than the one in
the root.the root.
 The new node is made the right child if theThe new node is made the right child if the
number is greater than the one in the root.number is greater than the one in the root.

9. Searching for DuplicatesSearching for Duplicates

10. Searching for DuplicatesSearching for Duplicates

11. Searching for DuplicatesSearching for Duplicates

12. Searching for DuplicatesSearching for Duplicates

13. Searching for DuplicatesSearching for Duplicates

14. Searching for DuplicatesSearching for Duplicates

15. Searching for DuplicatesSearching for Duplicates

16. Searching for DuplicatesSearching for Duplicates

17. Searching for DuplicatesSearching for Duplicates

22.  DuplicateDuplicate (4)(4) Appeared …..Appeared …..

26.  DuplicateDuplicate (9),(9), (14),(14), (5),(5), Appeared …..Appeared …..

27. Tree TraversalsTree Traversals
1.1. PreOrderPreOrder
2.2. InOrderInOrder
3.3. PostOrderPostOrder

28. PreorderPreorder
 To traverse non empty binary tree inTo traverse non empty binary tree in
PreorderPreorder,, we perform the followingwe perform the following
operations.operations.
1.1. Visit theVisit the RootRoot
2.2. Traverse theTraverse the leftleft sub tree insub tree in preorderpreorder
3.3. Traverse theTraverse the rightright sub tree insub tree in preorderpreorder

29. InorderInorder
 To traverse a non empty binary tree inTo traverse a non empty binary tree in
Inorder (or Symmetric Order)Inorder (or Symmetric Order),, wewe
perform the following operations.perform the following operations.
1.1. Traverse theTraverse the leftleft sub tree insub tree in InorderInorder
2.2. Visit theVisit the RootRoot
3.3. Traverse theTraverse the rightright sub tree insub tree in InorderInorder

30. PostorderPostorder
 To traverse non empty binary tree inTo traverse non empty binary tree in
PostorderPostorder,, we perform the followingwe perform the following
operations.operations.
1.1. Traverse theTraverse the leftleft sub tree insub tree in postorderpostorder
2.2. Traverse theTraverse the rightright sub tree insub tree in postorderpostorder
3.3. Visit theVisit the RootRoot

31. Example of Tree TraversalsExample of Tree Traversals
AA
CC
EE
HH
FF GG
DD
II
BB
JJ
LLKK
 PreOrder : A B C E I F J D G H K LPreOrder : A B C E I F J D G H K L
 InOrder : E I C F J B G D K H L AInOrder : E I C F J B G D K H L A
 PostOrder : I E J F C G K L H D B APostOrder : I E J F C G K L H D B A