binary tree power point presentation for iT

Introduction to Algorithms and
Data Structures
Lecture 12 - “I think that I shall never
see.. a data structure lovely as a”
Binary Tree

What is a Binary Tree
• A binary tree is a a collection of nodes that
consists of the root and other subsets to the root
points, which are called the left and right subtrees.
• Every parent node on a binary tree can have up to
two child nodes (roots of the two subtrees); any
more children and it becomes a general tree.
• A node that has no children is called a leaf node.

A Few Terms Regarding Binary Trees
A
E F
C
B
G
D
A is the root
B and C are A’s children
A is B’s parent
B & C are the root of two subtrees
D, E, and G are leaves

This is NOT A Binary Tree
A
E F
C
B
G
D H
This is a general tree because C has three child nodes

This is NOT A Binary Tree
A
E F
C
B
G
D
This is a graph
because A, B, E, H, F and C
form a circuit
H

Tree Traversal
• There are three common ways to traverse a tree:
– Preorder: Visit the root, traverse the left subtree
(preorder) and then traverse the right subtree (preorder)
– Postorder: Traverse the left subtree (postorder),
traverse the right subtree (postorder) and then visit the
root.
– Inorder: Traverse the left subtree (in order), visit the
root and the traverse the right subtree (in order).

Tree Traversals: An Example
A
E F
C
B
G
D
Preorder: ABDECFG
Postorder: DEBGFCA
In Order: DBEAFGC

A
E F
C
B
G
D
Preorder: A
(visit each node as your reach it)

A
E F
C
B
G
D
Preorder: AB

A
E F
C
B
G
D
Preorder: ABD

A
E F
C
B
G
D
Preorder: ABDE

A
E F
C
B
G
D
Preorder: ABDEC

A
E F
C
B
G
D
Preorder: ABDECF

A
E F
C
B
G
D
Preorder: ABDECFG

A
E F
C
B
G
D
Postorder:

A
E F
C
B
G
D
Postorder: D

A
E F
C
B
G
D
Postorder: DE

A
E F
C
B
G
D
Postorder: DEB

A
E F
C
B
G
D
Postorder: DEBG

A
E F
C
B
G
D
Postorder: DEBGF

A
E F
C
B
G
D
Postorder: DEBGFC

A
E F
C
B
G
D
Postorder: DEBGFCA

A
E F
C
B
G
D
In Order:

A
E F
C
B
G
D
In Order: D

A
E F
C
B
G
D
In Order: DB

A
E F
C
B
G
D
In Order: DBE

A
E F
C
B
G
D
In Order: DBEA

A
E F
C
B
G
D
In Order: DBEAF

A
E F
C
B
G
D
In Order: DBEAFG

A
E F
C
B
G
D
In Order: DBEAFGC

Tree Traversals: Another Example
A
E
F
C
B
I
D
Preorder: ABDFHIECG
Postorder: HIFDEBGCA
In Order: DHFIBEACG
G
H

A
E
F
C
B
I
D
Preorder: A
G
H

A
E
F
C
B
I
D
Preorder: AB
G
H

A
E
F
C
B
I
D
Preorder: ABD
G
H

A
E
F
C
B
I
D
Preorder: ABDF
G
H

A
E
F
C
B
I
D
Preorder: ABDFH
G
H

A
E
F
C
B
I
D
Preorder: ABDFHI
G
H

A
E
F
C
B
I
D
Preorder: ABDFHIE
G
H

A
E
F
C
B
I
D
Preorder: ABDFHIEC
G
H

A
E
F
C
B
I
D
Preorder: ABDFHIECG
G
H

A
E
F
C
B
I
D
Postorder:
G
H

A
E
F
C
B
I
D
Postorder: H
G
H

A
E
F
C
B
I
D
Postorder: HI
G
H

A
E
F
C
B
I
D
Postorder: HIF
G
H

A
E
F
C
B
I
D
Postorder: HIFD
G
H

A
E
F
C
B
I
D
Postorder: HIFDE
G
H

A
E
F
C
B
I
D
Postorder: HIFDEB
G
H

A
E
F
C
B
I
D
Postorder: HIFDEBG
G
H

A
E
F
C
B
I
D
Postorder: HIFDEBGC
G
H

A
E
F
C
B
I
D
Postorder: HIFDEBGCA
G
H

A
E
F
C
B
I
D
In Order:
G
H

A
E
F
C
B
I
D
In Order: D
G
H

A
E
F
C
B
I
D
In Order: DH
G
H

A
E
F
C
B
I
D
In Order: DHF
G
H

A
E
F
C
B
I
D
In Order: DHFI
G
H

A
E
F
C
B
I
D
In Order: DHFIB
G
H

A
E
F
C
B
I
D
In Order: DHFIBE
G
H

A
E
F
C
B
I
D
In Order: DHFIBEA
G
H

A
E
F
C
B
I
D
In Order: DHFIBEAC
G
H

A
E
F
C
B
I
D
In Order: DHFIBEACG
G
H

Basic Implementation of a Binary
Tree
• We can implement a binary in essentially
the same way as a linked list, except that
there are two nodes that comes next:
public class Node {
private int data;
private Node left;
private Node right;

public int getData() {
return data;
}
public Node getLeft() {
return left;
}
public Node getRight() {
return right;
}
public void setData(int x) {
data = x;
}

public void setLeft(Node p) {
left = p;
}
public void setRight(Node p) {
right = p;
}
}

The tree Class
public class Tree {
private Node root;
// tree() - The default constructor – Starts
// the tree as empty
public Tree() {
root = null;
}
// Tree() - An initializing constructor that
// creates a node and places in it
// the initial value

public Tree(int x) {
root = new Node();
root.setData(x);
root.setLeft(null);
root.setRight(null);
}
public Node getRoot() {
return root;
}

// newNode() - Creates a new node with a
// zero as data by default
public Node newNode() {
Node p = new Node();
p.setData(0);
p.setLeft(null);
p.setRight(null);
return(p);
}

// newNode() - Creates a new node with the
// parameter x as its value
public Node newNode(int x) {
Node p = new Node();
p.setData(x);
p.setLeft(null);
p.setRight(null);
return(p);
}

//travTree() – initializes recursive
// traversal of tree
public void travTree() {
if (root != null)
travSubtree(root);
System.out.println();
}
//travSubtree() – recursive method used to
// traverse a binary tree (inorder)
public void travSubtree(Node p) {
if (p != null) {
travSubtree(p.getLeft());
System.out.print(p.getData()
+ "t");
travSubtree(p.getRight());
}
}

// addLeft() - Inserts a new node containing
// x as the left child of p
public void addLeft(Node p, int x) {
Node q = newNode(x);
p.setLeft(q);
}
// addRight() - Inserts a new node containing
// x as the right child of p
public void addRight(Node p, int x) {
Node q = newNode(x);
p.setRight(q);
}
}

A Basic Search Tree
• We can construct a simple search tree if we
add new nodes with value x on the tree
using this strategy:
– Every time x is less than the value in the node
we move down to the left.
– Every time x is greater than the value in the
node we move down to the right.

A Sample Search Tree
15
12 18
22
8
20
5 23
16
10
7

// insert() - Insert value x in a new node to
// be inserted after p
public void insert(int x) {
Node p, q;
boolean found = false;
p = root;
q = null;
while (p != null && !found) {
q = p;
if (p.getData() == x)
found = true;
else if (p.getData() > x)
p = p.getLeft();
else
p = p.getRight();
}

if (found)
error("Duplicate entry");
if (q.getData() > x)
addLeft(q, x);
else
addRight(q, x);
//q = newNode(x);
}

// isXThere() - Is there a node in the
// tree containing x?
public boolean isXThere(int x) {
Node p;
p = root;
found = true;
p = p.getLeft();
else
p = p.getRight();
}
return(found);
}

public void error(String message) {
System.out.println(message);
System.exit(0);
}

// getNode() - Get the pointer for the
// node containing x
public Node getNode(int x) {
Node p, q;
p = root;
q = null;
q = p;
found = true;
p = p.getLeft();
else
p = p.getRight();
}

if (found)
return(q);
else
return(null);
}

public class TestTree {
public static void main(String[] args) {
Tree mytree = new Tree(8);
mytree.addLeft(mytree.getRoot(), 6);
mytree.addRight(mytree.getRoot(), 9);
mytree.insert(4);
mytree.insert(1);
mytree.insert(12);
if (mytree.isXThere(13))
System.out.println("great");
else
System.out.println("not great, Bob");
mytree.travTree();
}
}

binary tree power point presentation for iT

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