Bab 5 - Tree

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Data Structures Using Java 1

Bab 5

Binary Trees

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Chapter Objectives

Learn about binary trees

Explore various binary tree traversal algorithms

Learn how to organize data in a binary search tree

Discover how to insert and delete items in a binary

search tree

Explore nonrecursive binary tree traversalalgorithms

Learn about AVL (height-balanced) trees

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Binary Trees

Definition: A binary tree, T, is either empty or

such that:

Thas a special node called the rootnode; Thas two sets of nodes,LTandRT, called the left

subtree and right subtree ofT, respectively;

LTandRTare 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

For each node:

Data is stored in info

The reference to the left child is stored in llink The reference to the right child is stored in rlink

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General Binary Tree

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Binary Tree Definitions

Leaf: node that has no left and right children

Parent: node with at least one child node

Level of a node: number of branches on the pathfrom root to node

Height of a binary tree: number of nodes no 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

Useful operation on binary trees is to make

identical copy of binary tree

Method copy useful in implementing copyconstructor 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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Method copyTree

public void copyTree(BinaryTree otherTree)

{

if(this != otherTree) //avoid self-copy

{root = null;

if(otherTree.root != null) //otherTree is

//nonempty

root = copy(otherTree.root);

}}

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Binary Tree Traversal

Must start with the root, then

Visit the node first

or Visit the subtrees first

Three different traversals

Inorder

Preorder

Postorder

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Traversals

Inorder

Traverse the left subtree

Visit the node

Traverse the right subtree

Preorder

Visit the node

Traverse the left subtree Traverse the right subtree

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Traversals

Postorder

Traverse the left subtree

Traverse the right subtree 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){


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);


}

}

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Implementing Binary Trees:

class BinaryTree methods isEmpty

inorderTraversal

preorderTraversal postorderTraversal

treeHeight

treeNodeCount

treeLeavesCount destroyTree

copyTree

Copy

Inorder

Preorder postorder

Height

Max

nodeCount leavesCount

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Binary Search Trees

Data in each node

Larger than the data in its left child

Smaller than the data in its right child

A binary search tree, t, is either empty or:

Thas a special node called the rootnode

Thas two sets of nodes,LTandRT, called the leftsubtree and right subtree ofT, respectively

Key in root node larger than every key in left subtreeand smaller than every key in right subtree

LTandRTare binary search trees

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Binary Search Trees

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Operations Performed on Binary

Search Trees Determine whether the binary search tree is empty

Search the binary search tree for a particular item

Insert an item in the binary search tree

Delete an item from the binary search tree

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Operations Performed on Binary

Search Trees Find the height of the binary search tree

Find the number of nodes in the binary search tree

Find the number of leaves in the binary search tree

Traverse the binary search tree

Copy the binary search tree

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Binary Search Tree: Analysis

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Theorem: Let Tbe a binary search tree with n nodes,

where n > 0.The average number of nodes visited in a

search ofTis approximately 1.39log2n

Number of comparisons required to determine whetherx isin Tis one more than the number of comparisons required

to insertx in T

Number of comparisons required to insertx in T same as

the number of comparisons made in unsuccessful search,reflecting thatx is not in T

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It follows that:

It is also known that:

Solving Equations (10-1) and (10-2)

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Nonrecursive Inorder Traversal

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Nonrecursive Inorder Traversal:

General Algorithm1. 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 Algorithm1. 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;current = current.llink;

}

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;

current = current.llink;

}

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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;

}

else

visit current;

}

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AVL (Height-Balanced Trees)

A perfectly balancedbinary tree is a binary tree

such that:

The height of the left and right subtrees of the root areequal

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)

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

search tree such that:

The height of the left and right subtrees of the rootdiffer by at most 1

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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Insertion Into AVL Tree

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Insertion Into AVL Trees

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AVL Tree Rotations

Reconstruction procedure: rotatingtree

left rotationand right rotation

Suppose that the rotation occurs at nodex

Left rotation: certain nodes from the right subtree ofx

move to its left subtree; the root of the right subtree ofx

becomes the new root of the reconstructed subtree

Right rotation atx: certain nodes from the left subtree ofx

move to its right subtree; the root of the left subtree ofx

becomes 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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AVL Tree Rotations

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Deletion From AVL Trees

Case 1: the node to be deleted is a leaf

Case 2:the node to be deleted has no right child,

that is, its right subtree is empty Case 3:the node to be deleted has no left child,

that is, its left subtree is empty

Case 4:the node to be deleted has a left child and

a right child

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Analysis: AVL Trees

Consider all the possible AVL trees of height h. Let Thbe an

AVL tree of height h such that Thhas the fewest number of

nodes. Let Thldenote the left subtree ofT

hand T

hrdenote the

right subtree ofTh. Then:

where | Th| denotes the number of nodes in T

h.

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Analysis: AVL Trees

Suppose that Thlis of height h 1 and T

hris of height h 2.

Thlis an AVL tree of height h 1 such that T

hlhas the fewest

number of nodes among all AVL trees of height h 1. Thris

an AVL tree of height h 2 that has the fewest number ofnodes among all AVL trees of height h 2. T

hlis of the form

Th-1 and T

hris of the form T

h-2. Hence:

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Analysis: AVL Trees

LetFh+2 = |Th | + 1. Then:

Called a Fibonacci sequence; solution toFh

is given by:

Hence

From this it can be concluded that

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Programming Example: Video

Store (Revisited) In Chapter 4,we designed a program to help a video store automate its

video rental process.

That program used an (unordered) linked list to keep track of the videoinventory in the store.

Because the search algorithm on a linked list is sequential and the listis fairly large, the search could be time consuming.

If the binary tree is nicely constructed (that is, it is not linear), then thesearch algorithm can be improved considerably.

In general, item insertion and deletion in a binary search tree is faster

than in a linked list. We will redesign the video store program so that the video inventory

can be maintained in a binary tree.

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Chapter Summary

Binary trees

Binary search trees

Recursive traversal algorithms

Nonrecursive traversal algorithms

AVL trees

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