Transcript Document 7910858

Trees
EENG212
Algorithms
and
Data Structures
Trees
Outline
Introduction to Trees
 Binary Trees: Basic Definitions
 Traversing Binary Trees
 Node Representation of Binary Trees
 Primitive Functions in Binary Trees

Introduction to Trees
BINARY TREES: BASIC DEFINITIONS
 A binary tree is a finite set of elements that are
either empty or is partitioned into three disjoint
subsets. The first subset contains a single element
called the root of the tree. The other two subsets
are themselves binary trees called the left and right
subtrees of the original tree. A left or right subtree
can be empty.
 Each element of a binary tree is called a node of the
tree. The following figure shows a binary tree with 9
nodes where A is the root.
BINARY TREES: BASIC
DEFINITIONS
root
left subtree
right subtree
BINARY TREES: BASIC
DEFINITIONS

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If A is the root of a binary tree and B is the root of its
left or right subtrees, then A is said to be the father
of B and B is said to be the left son of A.
A node that has no sons is called the leaf.
Node n1 is the ancestor of node n2 if n1 is either
the father of n2 or the father of some ancestor of n2.
In such a case n2 is a descendant of n1.
Two nodes are brothers if they are left and right
sons of the same father.
BINARY TREES: BASIC
DEFINITIONS
left son
right son
leaves
BINARY TREES: BASIC
DEFINITIONS
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If every nonleaf node in a binary tree has
nonempty left and right subtrees, the tree is
called a strictly binary tree.
BINARY TREES: BASIC
DEFINITIONS
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The level of a node in a binary tree is defined
as follows: The root of the tree has level 0,
and the level of any other node in the tree is
one more than the level of its father.
The depth of a binary tree is the maximum
level of any leaf in the tree.
A complete binary tree of depth d is the
strictly binary all of whose leaves are at level
d. A complete binary tree with depth d has 2d
leaves and 2d-1 nonleaf nodes.
BINARY TREES: BASIC
DEFINITIONS
TRAVERSING BINARY TREES
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One of the common operations of a binary tree is to
traverse the tree. Traversing a tree is to pass
through all of its nodes once. You may want to print
the contents of each node or to process the
contents of the nodes. In either case each node of
the tree is visited.
There are three main traversal methods where
traversing a binary tree involves visiting the root and
traversing its left and right subtrees. The only
difference among these three methods is the order
in which these three operations are performed.
TRAVERSING BINARY TREES

Traversing a binary tree in preorder
(depth-first order)
1. Visit the root.
2. Traverse the left subtree in preorder.
3. Traverse the right subtree in preorder.
Traversing a binary tree in
preorder
Preorder: ABDGCEHIF
TRAVERSING BINARY TREES
Traversing a binary tree in inorder
(or symmetric order)
1. Traverse the left subtree in inorder.
2. Visit the root.
3. Traverse the right subtree in inorder.
Traversing a binary tree in
inorder
Inorder: DGBAHEICF
TRAVERSING BINARY TREES

Traversing a binary tree in postorder
1. Traverse the left subtree in postorder.
2. Traverse the right subtree in postorder.
3. Visit the root.
Traversing a binary tree in
postorder
Postorder: GDBHIEFCA
NODE REPRESENTATION OF
BINARY TREES

Each node in a binary tree contains info, left, right
and father fields. The left, right and father fields
points the node’s left son, right son and the father
respectively.
struct node{
int info; /* can be of different type*/
struct node *left;
struct node *right;
struct node *father;
};
typedef struct node *NODEPTR;
PRIMITIVE FUNCTIONS IN
BINARY TREES
The maketree function allocates a node and
sets it as the root of a single node binary tree.
NODEPTR maketree(int x)
{
NODEPTR p;
p = getnode();
p->info = x;
p->left = NULL;
p->right = NULL;
return p;
}
PRIMITIVE FUNCTIONS IN
BINARY TREES

The setleft and setright functions sets a node with
content x as the left son and right son of the node p
respectively.
void setleft(NODEPTR p, int x)
{
if(p == NULL){
printf(“void insertion\n”);
else if (p->left != NULL)
printf(“invalid insertion\n”);
else
p->left = maketree(x);
}
void setright(NODEPTR p, int x)
{
if(p == NULL){
printf(“void insertion\n”);
else if (p->right != NULL)
printf(“invalid insertion\n”);
else
p->right = maketree(x);
}
BINARY TREE TRAVERSAL
METHODS
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Recursive functions can be used to perform
traversal on a given binary tree. Assume that
dynamic node representation is used for a given
binary tree.
In the following traversal methods, the tree is
traversed always in downward directions. Therefore
the father field is not needed.
The following recursive preorder traversal function
displays the info part of the nodes in preorder. Note
that the info part is integer number and tree is a
pointer to the root of the tree.
BINARY TREE TRAVERSAL
METHODS
void pretrav(NODEPTR tree)
{
if(tree != NULL){
printf(“%d\n”, tree->info);
pretrav(tree->left);
pretrav(tree->right);
}
}
BINARY TREE TRAVERSAL
METHODS

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The following recursive inorder traversal
function displays the info part of the nodes in
inorder.
Note that the info part is integer number and
tree is a pointer to the root of the tree.
BINARY TREE TRAVERSAL
METHODS
void intrav(NODEPTR tree)
{
if(tree != NULL){
intrav(tree->left);
printf(“%d\n”, tree->info);
intrav(tree->right);
}
}
BINARY TREE TRAVERSAL
METHODS

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The following recursive postorder traversal
function displays the info part of the nodes in
postorder.
Note that the info part is integer number and
tree is a pointer to the root of the tree.
BINARY TREE TRAVERSAL
METHODS
void posttrav(NODEPTR tree)
{
if(tree != NULL){
posttrav(tree->left);
posttrav(tree->right);
printf(“%d\n”, tree->info);
}
}
BINARY SEARCH TREE: AN
APPLICATION OF BINARY TREES

A binary tree, that has the property that all
elements in the left subtree of a node n are
less than the contents of n, and all elements
in the right subtree of n are greater than or
equal to the contents of n, is called a Binary
Search Tree or Ordered Binary Tree.
BINARY SEARCH TREE: AN
APPLICATION OF BINARY TREES
Given the following sequence of numbers,
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
 The following binary search tree can be
constructed.

BINARY SEARCH TREE: AN
APPLICATION OF BINARY TREES
BINARY SEARCH TREE: AN
APPLICATION OF BINARY TREES
The inorder (left-root-right) traversal of the
above Binary Search Tree and printing the
info part of the nodes gives the sorted
sequence in ascending order. Therefore,
the Binary search tree approach can easily
be used to sort a given array of numbers.
 The inorder traversal on the above Binary
Search Tree is:
3, 4, 4, 5, 5, 7, 9, 9, 14, 14, 15, 16, 17, 18, 20

SEARCHING THROUGH THE
BINARY SEARCH TREE
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Searching operation of the binary search tree
is always in downward direction. Therefore
the following node structure can be used to
represent the node of a given binary search
tree.
Note that the father link is not required.
SEARCHING THROUGH THE
BINARY SEARCH TREE
struct node{
int info; /* can be of different type*/
struct node *left;
struct node *right;
};
typedef struct node *NODEPTR;
SEARCHING THROUGH THE
BINARY SEARCH TREE

The following recursive function can be used
to search for a given key element in a given
array of integers. The array elements are
stored in a binary search tree. Note that the
function returns TRUE (1) if the searched key
is a member of the array and FALSE (0) if the
searched key is not a member of the array.
SEARCHING THROUGH THE
BINARY SEARCH TREE
int BinSearch(NODEPTR p, int key)
{
if(p == NULL)
return FALSE;
else {
if (key == p->info)
return TRUE;
else{
if(key < p->info)
return BinSearch(p->left, key);
else
return BinSearch(p->right, key);
}
}
}
INSERTING NODES INTO A
BINARY SEARCH TREE

The following recursive function can be used
to insert a new node into a given binary
search tree.
NODEPTR insert(NODEPTR p, int x)
{
if(p == NULL){
p = getnode();
p->info = x;
p->left = NULL;
p->right = NULL;
return p;
}
else{
if(x < p->info)
p->left = insert(p->left, x);
else
p->right = insert(p->right, x);
return p;
}
}
Application of Binary Search
Tree

Suppose that we wanted to find all duplicates in a list of
numbers. One way of doing this to compare each number with all
those that precede it. However this involves a large number of
comparison. The number of comparison can be reduced by using
a binary tree. The first number in the list is placed in a node that
is the root of the binary tree with empty left and right sub-trees.
The other numbers in the list is than compared to the number in
the root. If it is matches, we have duplicate. If it is smaller, we
examine the left sub-tree; if it is larger we examine the right subtree. If the sub-tree is empty, the number is not a duplicate and is
placed into a new node at that position in the tree. If the sub-tree
is nonempty, we compare the number to the contents of the root
of the sub-tree and the entire process is repeated with the subtree. A program for doing this follows .
#include <stdio.h>
#include <stdlib.h>
struct node {
struct node *left ;
int info ;
struct node *right;
};
typedef struct node *NODEPTR;
NODEPTR maketree(int);
NODEPTR getnode(void);
void intrav(NODEPTR);
void main()
{
int number;
NODEPTR root , p , q;
printf("%s\n","Enter First number");
scanf("%d",&number);
root=maketree(number); /* insert first root item */
printf("%s\n","Enter the other numbers");
while(scanf("%d",&number) !=EOF)
{ p=q=root;
/* find insertion point */
while((number !=p->info) && q!=NULL)
{p=q;
if (number <p->info)
q = p->left;
else
q = p->right;
}
q=maketree(number);
/* insertion */
if (number==p->info)
printf("%d is a duplicate \n",number);
else if (number<p->info)
p->left=q;
else p->right=q;
}
printf("Tree Created \n ");
/* inorder Traversing */
intrav(root);
}
void intrav(NODEPTR tree)
{
if(tree != NULL){
intrav(tree->left);
printf(“%d\n”, tree->info);
intrav(tree->right);
}
}
NODEPTR maketree(int x)
{
NODEPTR p;
p = getnode();
p->info = x;
p->left = NULL;
p->right = NULL;
return p;
}
NODEPTR getnode(void)
{
NODEPTR p;
p=(NODEPTR) malloc(sizeof(struct node));
return p;
}