Prepare for your exams
Get points
Guidelines and tips
Sell on Docsity
Docsity AI

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Sell on Docsity

Docsity AI

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents

20 Points

For each uploaded document

Answer questions

5 Points

For each given answer (max 1 per day)

All the ways to get free points

Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Trees in data Structure, Lecture notes of Data Structures and Algorithms

AMET University Data Structures and Algorithms

Tree,Binary tree,Node,Linked Representation

Typology: Lecture notes

2018/2019

Uploaded on 06/26/2019

monaa_pin 🇮🇳

4 documents

1 / 110

This page cannot be seen from the preview

Don't miss anything!

Trees

•Definition: A tree is a finite set of

one or more nodes such that:

–There is a specially designated node

called the root.

–The remaining nodes are partitioned

into n ≥ 0 disjoint sets T1, …, Tn, where

each of these sets is a tree. We call T1,

…, Tn the subtrees of the root.

Discover Lecture notes of Data Structures and Algorithms AMET University

Partial preview of the text

Download Trees in data Structure and more Lecture notes Data Structures and Algorithms in PDF only on Docsity!

Trees

• Definition: A tree is a finite set of

one or more nodes such that:

– There is a specially designated node

called the root.

– The remaining nodes are partitioned

into n ≥ 0 disjoint sets T 1 , …, Tn, where

each of these sets is a tree. We call T 1 ,

…, Tn the subtrees of the root.

Pedigree Genealogical

Chart

Chery l Kevin Rosemar y John Terry Richard^ Kelly Jack (^) Mary Anthon y Karen Joe Michelle^ Mike Angela Binary Tree

Tree Terminology

• Normally we draw a tree with the root at the

top.

(^) The degree of a node is the number of

subtrees of the node.

(^) The degree of a tree is the maximum degree of

the nodes in the tree.

(^) A node with degree zero is a leaf or terminal

node.

(^) A node that has subtrees is the parent of the

roots of the subtrees, and the roots of the

subtrees are the children of the node.

(^) Children of the same parents are called

siblings.

Tree Terminology

(Cont.)

• The ancestors of a node are all the

nodes along the path from the root to the

node.

• The descendants of a node are all the

nodes that are in its subtrees.

• Assume the root is at level 1, then the

level of a node is the level of the node’s

parent plus one.

• The height or the depth of a tree is the

maximum level of any node in the tree.

List Representation of

Trees

A

B F 0 C G 0

D I J 0

E K L 0 H M 0

Possible Node Structure

For A Tree of Degree

(^) Lemma 5.1: If T is a k-ary tree (i.e., a tree of degree k) with n nodes, each having a fixed size as in Figure 5.4, then n(k-1) + 1 of the nk child fileds are 0, n ≥ 1. Data Child 1 Child 2 Child 3 Child 4 …^ Child k Wasting memory!

Degree Two Tree

Representation

A

B

C

D

E

F G

H

I

J

K

L

M

Binary Tree!

Binary Tree

• Definition: A binary tree is a finite set of

nodes that is either empty or consists of

a root and two disjoint binary trees

called the left subtree and the right

subtree.

• There is no tree with zero nodes. But

there is an empty binary tree.

• Binary tree distinguishes between the

order of the children while in a tree we

do not.

Binary Tree Examples

A

B

A

B

A

B

C

D

E

A

B C

D E F G

H I

The Properties of Binary

Trees

(^) Lemma 5.2 [Maximum number of nodes]

1) The maximum number of nodes on level i of a

binary tree is 2 i-1, i ≥ 1.

2) The maximum number of nodes in a binary tree of

depth k is 2 k^ – 1, k ≥ 1.

(^) Lemma 5.3 [Relation between number of

leaf nodes and nodes of degree 2]: For any

non-empty binary tree, T, if n 0 is the number

of leaf nodes and n2 the number of nodes of

degree 2, then n 0 = n 2 + 1.

(^) Definition : A full binary tree of depth k is

a binary tree of depth k having 2 k^ – 1 nodes,

k ≥ 0.

Binary Tree Definition

(^) Definition : A binary tree with n nodes and

depth k is complete iff its nodes correspond to

the nodes numbered from 1 to n in the full

binary tree of depth k.

leve l (^1) 2 3 4

Array Representation of A

Binary Tree

If a complete binary tree with n nodes is represented sequentially, then for any node with index i, 1 ≤ i ≤ n , we have:

(^) parent( i ) is at if i ≠1. If i = 1, i is at the

root and has no parent.

left_child( i ) is at 2 i if 2i ≤ n. If 2i > n, then i has no

left child.

(^) right_child( i ) is at 2 i + 1 if 2 i + 1 ≤ n. If 2 i + 1 > n ,

then i has no right child.

 i /^2 

Linked Representation

class Tree; class TreeNode { friend class Tree; private : TreeNode *LeftChild; char data; TreeNode *RightChild; }; class Tree { public : // Tree operations . private : TreeNode *root; };

Node Representation

LeftChild data RightChild data LeftChild RightChil d

Trees in data Structure, Lecture notes of Data Structures and Algorithms

Related documents

Partial preview of the text

Download Trees in data Structure and more Lecture notes Data Structures and Algorithms in PDF only on Docsity!

Trees

• Definition: A tree is a finite set of

one or more nodes such that:

– There is a specially designated node

called the root.

– The remaining nodes are partitioned

into n ≥ 0 disjoint sets T 1 , …, Tn, where

each of these sets is a tree. We call T 1 ,

…, Tn the subtrees of the root.

Pedigree Genealogical

Chart

Tree Terminology

• Normally we draw a tree with the root at the

top.

subtrees of the node.

the nodes in the tree.

node.

roots of the subtrees, and the roots of the

subtrees are the children of the node.

siblings.

Tree Terminology

(Cont.)

• The ancestors of a node are all the

nodes along the path from the root to the

node.

• The descendants of a node are all the

nodes that are in its subtrees.

• Assume the root is at level 1, then the

level of a node is the level of the node’s

parent plus one.

• The height or the depth of a tree is the

maximum level of any node in the tree.

List Representation of

Trees

A

B F 0 C G 0

D I J 0

E K L 0 H M 0

Possible Node Structure

For A Tree of Degree

Degree Two Tree

Representation

A

B

C

D

E

F G

H

I

J

K

L

M

Binary Tree

• Definition: A binary tree is a finite set of

nodes that is either empty or consists of

a root and two disjoint binary trees

called the left subtree and the right

subtree.

• There is no tree with zero nodes. But

there is an empty binary tree.

• Binary tree distinguishes between the

order of the children while in a tree we

do not.

Binary Tree Examples

A

B

A

B

A

B

C

D

E

A