1. Tree: A Non-linear Data Structure
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2. Background
• Linked lists use dynamic memory allocation,
and hence enjoys some advantages over static
representations using arrays.
• A problem
– Searching an ordinary linked list for a given
element.
• Time complexity O(n).
– Can the nodes in a linked list be rearranged so that
we can search in O(nlog2n) time?
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3. Illustration
50 Level 1
30 75 Level 2
12 45 53 80 Level 3
20 35 48 78 85 Level 4
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4. Binary Tree
• A new data structure
– Binary means two.
• Definition
– A binary tree is either empty, or it consists of a node
called the root, together with two binary trees called
the left subtree and the right subtree.
Root
Left Right
subtree subtree
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5. Examples of Binary Trees
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6. Not Binary Trees
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7. Binary Search Trees
• A particular form of binary tree suitable for
searching.
• Definition
– A binary search tree is a binary tree that is either
empty or in which each node contains a key that
satisfies the following conditions:
• All keys (if any) in the left subtree of the root precede the
key in the root.
• The key in the root precedes all keys (if any) in its right
subtree.
• The left and right subtrees of the root are again binary
search trees.
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8. Examples
10 50
5 15 30 75
12 20
12 45 53 80
20 35 48 78 85
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9. How to Implement a Binary Tree?
• Two pointers in every node (left and right).
struct nd {
int element;
struct nd *lptr;
struct nd *rptr;
};
typedef nd node;
node *root; /* Will point to the root of the tree */
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10. An Example
a
• Create the tree 10
b 5 20 c
15 d
a = (node *) malloc (sizeof (node));
b = (node *) malloc (sizeof (node));
c = (node *) malloc (sizeof (node));
d = (node *) malloc (sizeof (node));
a->element = 10; a->lptr = b; a->rptr = c;
b->element = 5; b->lptr = NULL; b->rptr = NULL;
c->element = 20; c->lptr = d; c->rptr = NULL;
d->element = 15; d->lptr = NULL; d->rptr = NULL;
root = a;
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11. Traversal of Binary Trees
• In many applications, it is required to move
through all the nodes of a binary tree, visiting
each node in turn.
– For n nodes, there exists n! different orders.
– Three traversal orders are most common:
• Inorder traversal
• Preorder traversal
• Postorder traversal
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12. Inorder Traversal
• Recursively, perform the following three steps:
– Visit the left subtree.
– Visit the root.
– Visit the right subtree.
LEFT-ROOT-RIGHT
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13. Example:: inorder traversal
10
10 20 30
20 30 40 25 50 60
20 10 30 40 20 25 10 50 30 60
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14. a
b c
d e f
g h i
j k
. d g b . a h e j i k c f
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15. Preorder Traversal
• Recursively, perform the following three steps:
– Visit the root.
– Visit the left subtree.
– Visit the right subtree.
ROOT-LEFT-RIGHT
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16. Example:: preorder traversal
10
10 20 30
20 30 40 25 50 60
10 20 30 10 20 40 25 30 50 60
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17. a
b c
d e f
g h i
j k
a b d . g . c e h i j k f
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18. Postorder Traversal
• Recursively, perform the following three steps:
– Visit the left subtree.
– Visit the right subtree.
– Visit the root.
LEFT-RIGHT-ROOT
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19. Example:: postorder traversal
10
10 20 30
20 30 40 25 50 60
20 30 10 40 25 20 50 60 30 10
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20. a
b c
d e f
g h i
j k
. g d . b h j k i e f c a
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21. Implementations
void inorder (node *root) void preorder (node *root)
{ {
if (root != NULL) if (root != NULL)
{ {
inorder (root->left); printf (“%d “, root->element);
printf (“%d “, root->element); inorder (root->left);
inorder (root->right); inorder (root->right);
} }
} }
void postorder (node *root)
{
if (root != NULL)
{
inorder (root->left);
inorder (root->right);
printf (“%d “, root->element);
}
}
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22. A case study :: Expression Tree
*
+ - Represents the expression:
(a + b) * (c – (d + e))
a b c +
d e
Preorder traversal :: * + a b - c + d e
Postorder traversal :: a b + c d e + - *
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23. Binary Search Tree (Revisited)
• Given a binary search tree, how to write a program to
search for a given element?
• Easy to write a recursive program.
int search (node *root, int key)
{
if (root != NULL)
{
if (root->element == key) return (1);
else if (root->element > key)
search (root->lptr, key);
else
search (root->rptr, key);
}
}
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24. Some Points to Note
• The following operations are a little complex
and are not discussed here.
– Inserting a node into a binary search tree.
– Deleting a node from a binary search tree.
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25. Graph :: another important data structure
• Definition
– A graph G={V,E} consists of a set of vertices V, and
a set of edges E which connect pairs of vertices.
– If the edges are undirected, G is called an
undirected graph.
– If at least one edge in G is directed, it is called a
directed graph.
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26. How to represent a graph in a program?
• Many ways
– Adjacency matrix
– Incidence matrix, etc.
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27. Adjacency Matrix
e1 e5
1 2 6
e2 e4 e6 e7
e3 e8
3 4 5
. 1 2 3 4 5 6
1 0 1 1 0 0 0
2 1 0 0 1 1 1
3 1 0 0 1 0 0
4 0 1 1 0 1 0
5 0 1 0 1 0 1
6 0 1 0 0 1 0
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28. Incidence Matrix
e1 e5
1 2 6
e2 e4 e6 e7
e3 e8
3 4 5
. e1 e2 e3 e4 e5 e6 e7 e8
1 1 1 0 0 0 0 0 0
2 1 0 0 1 1 1 0 0
3 0 1 1 0 0 0 0 0
4 0 0 1 1 0 0 0 1
5 0 0 0 0 0 1 1 1
6 0 0 0 0 1 0 1 0
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29. January 5, 2013 Programming and 29