Ceng-112 Data Structures I 1 Chapter 7 Introduction to Trees.

Ceng-112 Data Structures I 1

Chapter 7

IntroductiontoTrees


Introduction to Trees

• In the middle of nineteenth century, Gustav Kirchhoff studied

on trees in mathematics.

• Several years later, Arthur Cayley used them to study the

structure of algebraic formulas.

• In 1951, Grace Hopper’s use of them to represent arithmetic

expressions.

• Hopper’s work bears a strong resemblance today’s binary tree

formats.


Introduction to Trees

• Trees are used in computer science;

– To represent algebraic formulas,

– As an efficient method for searching large dynamic lists,

– For such diverse applications as artificial intelligence

systems,

– And encoding algorithms.


Basic Concepts

• A tree consists of a finite set of elements, called nodes.

• A tree consists of a finite set of directed lines, called branches.

These braches connect the nodes.

• The branch is directed towards the node, it is an indegree branch.

• The branch is directed away from the node, it is an outdegree

branch.


Figure 7-1

• The sum of the indegree and outdegree branches equals the degree

of the node.• If the tree is not empty, then the first node is called the root.

The indegree of the root is zero.• All of the nodes in a tree (exception of root) must have an indegree

of exactly one.

Basic Concepts


Basic Concepts

Leaf,Outdegree= 0

Internal Nodes,is not a root or a leaf.

Parent,Outdegree > 0

Child,indegree > 0

Siblings, with the same parent.

• Ancestor, is any node in the path

from the root node.• Descendent is,

all nodes in the path from givennode to the leaf.


Figure 7-2

Basic Concepts

Height of tree = max. Level of leaf + 1


Figure 7-3

• A subtree is any connected structure below the root.• A subtree can be divided into subtrees.

Basic Concepts


algorithm ConvertToParent(val root <node pointer>, ref output <string>)

Convert a general tree to parenthetical notation.Pre root is a pointer to a tree node.Post output contains parenthetical notation.1. Place root in output2. If (root is parent)

1. Place an open parenthesis in the output2. ConvertToParent(root’s first child)3. While (more siblings)

• ConvertToParent(root’s next child)4. Place close parenthesis in the output

3. ReturnEnd ConvertToParent

Root ( B ( C D ) E F ( G H I) )


Figure 7-5

Binary Trees

A binary tree is a tree in which no node can have more than two subtrees.


Figure 7-6

Null tree

Symmetry isnot a treerequirement!

Binary Trees


• Maximum height of tree for N nodes:

Hmax = N

• The minimum height of the tree :

Hmin = [log2N] + 1

• If known the height of a tree:

Nmin = H

Nmax = 2H-1

Binary Trees


Figure 7-7

Binary Trees - Balance

•A complete tree has the maximum number of entries for its heigh. Nmax = 2H-1

•The distance of a node from the root determines how efficiently it can be located.

• The “balance factor” show that the balance of the tree.

B = HL – HR

• If B = 0, 1 or -1; the tree is balanced.


Binary Tree Structure

• Each node in the structure must contain the data to be stored

and two pointers, one to the left subtree and one to the right

subtree.

Node

leftSubTree <pointer to Node>

data <dataType>

rightSubTree <pointer to Node>

End Node


Figure 7-25


Figure 7-8

Binary Tree Traversals

• A binary tree travelsal requires each node of the tree be processed once.

• In the depth-first traversal, all of the descendents of a child are processed before the

next child.

• In the breadth-first traversal, each level is completely processed before the next level

is started.

Three different depth-first traversal sequences.

NLR LNR LRN


Figure 7-9

Binary Tree Traversals

Preorder = ?Inorder = ?Postorder = ?


Figure 7-10

Binary Tree Traversals - Preorder


Figure 7-11Recursive algorithmic traversal of binary tree.

Binary Tree Traversals - Preorder


Figure 7-12

Binary Tree Traversals - Inorder


Figure 7-13

Binary Tree Traversals - Postorder


Figure 7-14

Binary Tree – Breadth-First Traversals


Figure 7-15

Expression Trees

An expression tree is a binary tree with these properties:

1. Each leaf is an operand.

2. The root and internal nodes are operators.

3. Subtrees are subexpressions with the root being an

operator.


Figure 7-16

Infix Traversal Of An Expression Tree


Infix Traversal Of An Expression Tree

algorithm infix (val tree <tree pointer>)Print the infix expression for an expression tree.Pre tree is a pointer to an expression treePost the infix expression has been printed1. If (tree not empty)

1. if (tree->token is an operand)1. print (tree->token)

2 else 1. print (open parenthesis)2. infix(tree->left)3. print(tree->token)4. infix(tree->right)5. print(close parenthesis)

2. Returnend infix


Huffman Code

• ASCII: 7 bits each character• Some characters occur more often than others, like 'E'• Every character uses the maximum number of bits• Huffman, makes it more efficient

– Assign shorter codes to ones that occur often– Longer codes for the ones that occur less often

• Typical frequencies:


Huffman Code

1. Organize character set into a row, ordered by frequency.

2. Find two nodes with smallest combined weight, join them and form a third.

3. Repeat until ALL are combined into a tree..


Huffman...


Huffman


Huffman...

• Now we assign a code to each character• Assign bit value for each branch:

– 0 = left branch,– 1 = right branch.

• A character's code is found by starting at root and following the branches.


Huffman

•

Note that the letters that occur most often are represented with very few bits


General Trees

• A general tree is a tree which each node can have an unlimited

outdegree.

• Binary trees are presented easier then general trees in

programs.

• In general tree, there are two releationships that we can use:

– Parent to child and,

– Sibling to sibling.

Using these two relationships, we can represent any general tree

as a binary tree.


Figure 7-17

Converting General Trees To Binary Trees


Figure 7-18

Insertion Into General Trees

FIFO insertion; the nodes are inserted at the end of the sibling list, (like insertion at the rear of the queue).


Figure 7-19

LIFO insertion; places the new node at the beginning of the sibling list, (like insertion at the front of the stack).



Figure 7-20


Key-sequence insertion; places the new node in key sequence amongthe sibling nodes.


Excercises

• Show the tree representation of the following parenthetical notation:

a ( b c ( e (f g) ) h )


Figure 7-21

Find:1. Root2. Leaves3. Internal nodes4. Ancestors of H5. Descendents of F6. Indegree of F7. Outdegree of B8. Level of G9. Heigh of node I

Excercises


Figure 7-22

Excercises

What is the balance factor of the below tree?


Figure 7-23

Exercises

Find the infix, prefix and postfix expressions of the below tree.


Exercise (Quiz?!)

Write the binary tree preorder traversal algorithm using a stack instead of recursion.

Algorithm preorderTraverse(ref tree <pointer>, stack <pointer of stack>)

Pre: tree variable has the address of the non-empty binary tree.

stack variable has the address of an empty stack.

Post: Binary tree is printed in preorder sequence.

1. initialize address variable with the pointer value of the binary tree

2. Push (stack, tree)

3. while (stack is not empty)

1.address=Pop(stack)

2.write(address->value)

3.if (address->right != null)

• Push(stack, address->right

4.if (address->left != null)

1. Push(stack, address->left)

4. end

A

B E

C D F G

addressABCDE

Printed: A,B,C,D,E,F,GA

E

B

D

C

G

F


HW-7

Write a program to:

• Create the following binary tree and;

• Create a menu to select the printing of infix, prefix and postfix expressions of the tree.

• Print the tree selected expression type.

Load your HW-6 to FTP site until 04 May. 07 at 09:00 am.

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Ceng-112 Data Structures I 1 Chapter 7 Introduction to Trees.

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