DSchap10

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Binary Trees

1

Chapter 10

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Chapter Objectives

Data Structures Using Java2

y Learn about binary trees

y Explore various binary tree traversal algorithms

y Learn how to organize data in a binary search tree

y Discover how to insert and delete items in a binary searchtree

y Explore nonrecursive binary tree traversal algorithms

y Learn about AVL (height-balanced) trees

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Binary Trees


y Definition: A binary tree, T, is either empty or such that:

y Thas a special node called the root node;

y Thas two sets of nodes, LTand RT, called the left subtree and

right subtree ofT, respectively;

y LTand RTare binary trees

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Binary Tree


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Binary Tree with One Node


The root node of the binary tree = A

LA = empty

RA = empty

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Binary Tree with Two Nodes


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Binary Tree with Two Nodes


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Various Binary Trees with Three Nodes


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Binary Trees


Following class defines the node of a binary tree:

protected class BinaryTreeNode

{

DataElement info;BinaryTreeNode llink;

BinaryTreeNode rlink;

}

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Nodes


y For each node:

y Data is stored in info

y The reference to the left child is stored in llink

y

The reference to the right child is stored in rlink

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General Binary Tree


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Binary Tree Definitions


y Leaf: node that has no left and right children

y Parent: node with at least one child node

y Level of a node: number of branches on the path from root to

nodey Height of a binary tree: number of nodes on the longest path

from root to node

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Height of a Binary Tree


Recursive algorithm to find height of binary

tree: (height(p) denotes height of binary tree

with root p):

if(p is NULL)

height(p) = 0

else

height(p) = 1 + max(height(p.llink),height(p.rlink))

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Height of a Binary Tree


Method to implement above algorithm:

private int height(BinaryTreeNode p)

{if(p == NULL)

return 0;

else

return 1 + max(height(p.llink),

height(p.rlink));}

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Copy Tree


y Useful operation on binary trees is to make identical copy of

binary tree

y Method copy useful in implementing copy constructor and

method copyTree

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Method copy


BinaryTreeNode copy(BinaryTreeNode otherTreeRoot)

{

BinaryTreeNode temp;

if(otherTreeRoot == null)

temp = null;

else

{

temp = new BinaryTreeNode();

temp.info = otherTreeRoot.info.getCopy();

temp.llink = copy(otherTreeRoot.llink);

temp.rlink = copy(otherTreeRoot.rlink);

}

return temp;

}//end copy

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Binary Tree Traversal


y Must start with the root, then

y Visit the node first

or

y

Visit the subtrees firsty Three different traversals

y Inorder

y Preorder

y

Postorder

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Traversals


y Inorder

y Traverse the left subtree

y Visit the node

y Traverse the right subtree

y Preordery Visit the node



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Traversals


y Postorder



y

Visit the node

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Binary Tree: Inorder Traversal


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Binary Tree: Inorder Traversal


private void inorder(BinaryTreeNode p)

{

if(p != NULL){

inorder(p.llink);

System.out.println(p.info + );

inorder(p.rlink);}

}

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Binary Tree: Preorder Traversal


private void preorder(BinaryTreeNode p)

{

if(p != NULL){

System.out.println(p.info + );

preorder(p.llink);

preorder(p.rlink);}

}

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Binary Tree: Postorder Traversal


private void postorder(BinaryTreeNode p)

{

if(p != NULL){

postorder(p.llink);

postorder(p.rlink);

System.out.println(p.info + );}

}

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ImplementingBinary Trees:

class BinaryTree methods


y isEmpty

y inorderTraversal

ypreorderTraversal

y postorderTraversal

y treeHeight

y treeNodeCount

y treeLeavesCount

y destroyTree

y copyTree

Copy

Inorder

Preorder postorder

Height

Max

nodeCount

leavesCount

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Binary Search Trees


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Operations Performed on Binary

Search Trees


y Determine whether the binary search tree is empty

y Search the binary search tree for a particular item

y Insert an item in the binary search tree

y Delete an item from the binary search tree

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Operations Performed on Binary

Search Trees


y Find the height of the binary search tree

y Find the number of nodes in the binary search tree

y Find the number of leaves in the binary search tree

y Traverse the binary search tree

y Copy the binary search tree

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Nonrecursive Inorder Traversal


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Nonrecursive Inorder Traversal:

General Algorithm


1. current = root; //start traversing the binary tree at

// the root node

2. while(current is not NULL or stack is nonempty)

if(current is not NULL)

{

push current onto stack;

current = current.llink;

}

else

{

pop stack into current;

visit current; //visit the node

current = current.rlink; //move to the right child}

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Nonrecursive Preorder Traversal

General Algorithm


1. current = root; //start the traversal at the root node

2. while(current is not NULL or stack is nonempty)

if(current is not NULL)

{

visit current;

push current onto stack;


}

else

{

pop stack into current;

current = current.rlink; //prepare to visit

//the right subtree}

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Nonrecursive Postorder Traversal


1. current = root; //start traversal at root node

2. v = 0;

3. if(current is NULL)

the binary tree is empty

4. if(current is not NULL)

a. push current into stack;

b. push 1 onto stack;

c. current = current.llink;

d. while(stack is not empty)

if(current is not NULL and v is 0)

{

push current and 1 onto stack;


}

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Nonrecursive Postorder Traversal

(Continued)


else

{

pop stack into current and v;

if(v == 1){

push current and 2 onto stack;

current = current.rlink;

v = 0;

}

elsevisit current;

}

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Perfect-Balanced Trees)


y A perfectly balanced binary tree is a binary tree such that:

y The height of the left and right subtrees of the root are equal

y The left and right subtrees of the root are perfectly balanced

binary trees

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Perfectly Balanced Binary Tree


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AVL (Height-Balanced Trees)


y An AVL tree (or height-balanced tree) is a binary search tree

such that:

y The height of the left and right subtrees of the root differ by at

most 1

y The left and right subtrees of the root are AVL trees

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AVL Trees


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Non-AVL Trees


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Balance Factor (bf)


y The balance factor of x, bf(x) is defined as:

bf(x) = xr xl

if x is left high, bf(x) = -1if x is equal high, bf(x) = 0

if x is right high, bf(x) = 1

y x violates the balance criteria if |xr xl| > 1

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Insertion Into AVL Tree


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Insertion Into AVL Trees


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AVL Tree Rotations


y Reconstruction procedure: rotating tree

y left rotation and right rotation

y Suppose that the rotation occurs at node x

y Left rotation: certain nodes from the right subtree ofxmove to its

left subtree; the root of the right subtree ofxbecomes the new

root of the reconstructed subtree

y Right rotation at x: certain nodes from the left subtree ofxmove

to its right subtree; the root of the left subtree ofxbecomes the

new root of the reconstructed subtree

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AVL Tree Rotations


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AVL Tree Rotations


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AVL Tree Rotations


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AVL Tree Rotations


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AVL Tree Rotations


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