This document discusses binary trees and their properties. It begins with defining key terms like root, parent, child, internal and external nodes. It then explains the differences between linear and non-linear data structures, providing examples of each. The document goes on to cover binary tree representations, properties such as balance factors, and applications of binary trees like search engines and game AI. It concludes by listing references for further reading.
Common Use of Tree as a Data Structure
ADVANCE
1. Nodes
2. Parent Nodes & Child Nodes
3. Leaf Nodes
4. Root Node
5. Sub Tree
6. Level of a tree:
7. m-ary Tree
8. Binary Tree (BT)
9. Complete and Full Binary Tree
10. Traversal
11. Binary Search Tree (BST)
12. Inorder Traversal – Left_ParentNode_Right
13. Postorder Traversal – Left_Right_ParentNode
14. Preorder Traversal – ParentNode_Left_Right
15. Binary Search Tree (BST)
16. BST - Insert, Delete
Students can learn Trees concept in data structures. various types of data structures like binary trees, expression trees, binary search trees and AVL trees are covered in this PPT.
Depth First Search Traversal of Binary Tree - Recursive and Non-recursive In-order, preorder, post-order traversal, Breath First Traversal of Binary Tree
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers introduction to paging in 80386, Address Translation (Linear to physical), Page Level Protection,
Common Use of Tree as a Data Structure
ADVANCE
1. Nodes
2. Parent Nodes & Child Nodes
3. Leaf Nodes
4. Root Node
5. Sub Tree
6. Level of a tree:
7. m-ary Tree
8. Binary Tree (BT)
9. Complete and Full Binary Tree
10. Traversal
11. Binary Search Tree (BST)
12. Inorder Traversal – Left_ParentNode_Right
13. Postorder Traversal – Left_Right_ParentNode
14. Preorder Traversal – ParentNode_Left_Right
15. Binary Search Tree (BST)
16. BST - Insert, Delete
Students can learn Trees concept in data structures. various types of data structures like binary trees, expression trees, binary search trees and AVL trees are covered in this PPT.
Depth First Search Traversal of Binary Tree - Recursive and Non-recursive In-order, preorder, post-order traversal, Breath First Traversal of Binary Tree
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers introduction to paging in 80386, Address Translation (Linear to physical), Page Level Protection,
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Introduction to multitasking, Support Registers and Data Structures, Task State Segment (TSS), TSS Descriptor, Task Register, Task Switching via TSS and Task Gate, Task Gate Descriptor,
PAI Unit 2 Segmentation in 80386 microprocessorKanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers types of address spaces : Logical, linear, Physical, Address Translation in 80386, Segment Descriptor Format, Types of Segment Descriptors,
PAI Unit 2 Protection in 80386 segmentationKanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers protection mechanism in 80386 microprocessor through conforming code segment and call gate
SE PAI Unit 2_Data Structures in 80386 segmentationKanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Descriptor Tables in 80386 as Global Descriptor Table, Local Descriptor Table, Types of Interrupts/Exception : Traps, faults, Aborts, Real mode Interrupt Structure (IVT), Protected mode interrupt Structure (IDT)
SE PAI Unit 5_Timer Programming in 8051 microcontroller_Part 1KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Introduction to Timers, Special Function Registers as Timer 1 Register, Timer 0 Register, TMOD register, TCON register, Operating modes of Timer
SE PAI Unit 5_Timer Programming in 8051 microcontroller_Part 2KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Mode 1 and Mode 2 programming of timers, Counters and Counter Programming
SE PAI Unit 5_Serial Port Programming in 8051 micro controller_Part 3KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers importance of RI Flag, Importance of TI flag, examples on Serial Port programming
SE PAI Unit 5_Serial Port Programming in 8051 microcontroller_Part 2KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers special function registers used for serial communication in 8051, Operating modes of serial communication, doubling baud rate in 8051
SE PAI Unit 5_Serial Port Programming in 8051 microcontroller_Part 1KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers the basics of serial communication, Data framing and Baud Rate in 8051 microcontroller.
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers ports of 8051 microcontroller, Description and How to configure those ports with examples
Unit 5_Interrupt programming in 8051 micro controller - part 2KanchanPatil34
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Special Function Registers of Interrupt programming of 8051 (IE, IP registers), example on interrupt programming
2015 course SPPU SEIT syllabus of subject Processor Architecture and Interfacing (PAI) This covers Introduction to Interrupts in 8051, Interrupt Handler, Types of Interrupts etc.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
1. Data Structures
Sanjivani Rural Education Society’s
Sanjivani College of Engineering, Kopargaon-423603
(An Autonomous Institute Affiliated to Savitribai Phule Pune University, Pune)
NAAC ‘A’ Grade Accredited, ISO 9001:2015 Certified
Department of Information Technology
(NBAAccredited)
Ms. K. D. Patil
Assistant Professor
2. Tree
• General Trees, Tree Terminology, Binary Trees, Use binary trees,
Conversion of general tree to binary tree, Array Based
representation of Binary Tree, Binary tree as an ADT, Binary tree
traversals - recursive and non-recursive algorithms, Construction
of tree from its traversals, Huffman coding algorithm
Data Structures Mrs. Kanchan Patil Department of Information Technology
3. Tree
• Linear Vs. Non-linear Data Structure
• Introduction to Tree
• Basic Tree Terminologies
• Example
Data Structures Mrs. Kanchan Patil Department of Information Technology
4. Learning Outcomes
• At the end of this lecture, student will able to-
• Differentiate linear and Non-linear Data Structures
• Introduce Tree data structure
• Explain different terms related to Tree data structure
Data Structures Mrs. Kanchan Patil Department of Information Technology
5. Recap!!!
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Data Structures: It is a particular way of organizing data in a computer.
Reference https://www.geeksforgeeks.org/difference-between-linear-and-non-linear-data-structures/
6. Linear Vs. Non-Linear Data Structures
Data Structures Mrs. Kanchan Patil Department of Information Technology
Sr. No. Linear Data Structure Non-Linear Data Structure
1 Data elements are arranged in a linear order. Elements
are attached to its previous and next adjacent.
Data elements are attached in hierarchically manner.
2 Single level is involved. Multiple levels are involved.
3 Its implementation is easy in comparison to non-linear
data structure.
Its implementation is complex in comparison to linear
data structure.
4 Data elements can be traversed in a single run only. Data elements can’t be traversed in a single run only.
5 Memory is not utilized in an efficient way. Memory is utilized in an efficient way.
6 Examples: array, stack, queue, linked list, etc. Examples: trees and graphs.
7 Applications:
Stack: undo, redo functions in editor
Queue: CPU job scheduling
Arrays: Storage of matrices
Linked List: Polynomial implementation for
mathematical operations
Applications: In Artificial Intelligence and image
processing.
Tree: 1. Implementing the hierarchical structures in
computer systems like directory and file system.
2. Implementing the navigation structure of a website.
Graph: Routes in GPS
7. Introduction to Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Non-linear data structure
• It has a structure that forms a parent-child relationship
• Collection of nodes and edges, the edges could be directed or undirected
• Definition (recursively):
• A tree is a finite set of one or more nodes such that
• There is a specially designated node called root.
• The remaining nodes are partitioned into n>=0 disjoint set T1,…,Tn, where
each of these sets is a tree. T1,…,Tn are called the subtrees of the root.
• Every node in the tree is the root of some subtree
9. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Node: It is a separate data structure that contains data or
value for a tree.
• Root: It is the top node in a tree. In any tree, first node is root
node.
• Parent: A node that has sub trees is the parent of the roots of
the sub-trees. The node which is a predecessor of any node is
called as Parent Node. The node which has a branch from it
to any other node is called a parent node. Parent node can
also be defined as "The node which has child / children".
10. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Children: A node connected to the previous node is known as
the child node. Each node in the tree is the child node of the
root node. The node which is descendant of any node is called
as Child Node. The node which has a link from its parent node is
called as child node. In a tree, any parent node can have any
number of child nodes.
• Siblings: Children of the same parent are siblings. The nodes
with the same parent are called Sibling nodes.
• Size: The total number of nodes present in the tree.
11. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Ancestors: All the nodes along the path from the root to the
node.
• Edge: The connecting link between any two nodes is called
as Edge. In a tree with 'N' number of nodes there will be a
maximum of 'N-1' number of edges.
• Terminal nodes (or leaf): The node which does not have a
child is called as leaf Node. Leaf has degree 0. A leaf is a
node with no child. The leaf nodes are also called
as External Nodes/Terminal Nodes.
12. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Non-terminal nodes: An internal node is a node with atleast
one child. Nodes other than leaf nodes are called as Internal
Nodes/ Non-terminal nodes. The root node is also said to be
Internal Node if the tree has more than one node.
• Degree of a node: The degree of a node is the number of sub
trees of the node. The total number of children of a node is
called as Degree of that Node.
• Degree of a tree: The highest degree of a node among all the
nodes in a tree is called as 'Degree of Tree'
13. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Level of Node: If a node is at level l, then it children are at level l+1. if, the root
node is said to be at Level 0 then children of root node are at Level 1 and the
children of the nodes which are at Level 1 will be at Level 2 and so on. Each step
from top to bottom is called as a Level and the Level count starts with '0' and
incremented by one at each level.
14. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Depth/ Height:
• The height of tree is the level of the leaf in the longest path from the root plus 1
• So, By definition,
• Height of empty tree is -1
• Example:
• Height of given tree = 3+1 =4
15. Basic Tree Terminologies
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Path: The sequence of Nodes and Edges from one
node to another node is called as path between that
two Nodes. Length of a Path is total number of nodes
in that path. In given example, the path A - B - E - J has
length 4.
• Sub-tree: A tree T is a tree consisting of a node
in T and all of its descendants in T. Each child from a
node forms a subtree recursively. Every child node will
form a subtree on its parent node.
17. Quiz…
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Root Node: A
• Parent of D and E: B
• sibling of B: C
• children of B: D and E
• External nodes : D, E, F, G, I
• Internal Nodes : A, B, C, H
• The level of E: 2
• The height (depth) of the tree: 4
• The degree of node B: 2
• The degree of the tree: 3
• The ancestors of node I: A, C, H
• The descendants of node C: F, G, H, I
18. Tree Representation
Data Structures Mrs. Kanchan Patil Department of Information Technology
• A tree is generally implemented in computer using “pointers/links”
• But, outside the computer systems, it can be represented using
• Organization chart format
• Indented list
• Parenthetical listing
19. Tree Representation
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Organization chart format
• Notations we use to represent tree is called
general tree
• Not easily implemented in computer systems
database
• Large structure may get complex
• Example:
• Computer parts list as a general tree
Computer
CPU
Controller ALU
Hardware
devices
Pen-Drive CD-ROM
20. Tree Representation
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Indented list
• Represents the assembly structure of an item
• Overcomes the disadvantage of chart format
• Sometimes, called as “goezinto” means “goes
into”
• Example:
• Computer parts list as a indented list
3. Computer
3.1 CPU
3.1.1 Controller
3.1.2 ALU
3.2 Hardware Devices
3.2.1 CD-ROM
3.2.2 Pen-Drive
21. Tree Representation
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Parenthetical listing
• It is used in algebraic expressions
• When a tree is represented in
parenthetical notations, each open
parenthesis indicates the start of new
level
• Each closing parenthesis completes
the current level and moves up one
level in the tree
• Example
• Parenthetical notation for below
tree is: A(B(D(HI)E(J))C(FG(K)))
22. Introduction to Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• A tree in which every node can have a maximum of two children is called as Binary
Tree
• It is a tree in which no node can have more than two sub-trees
• Every node can have either 0 children or 1 child or 2 children but not more than 2
children
• The sub-trees are designated as left sub-tree (left child) and the other one is right
sub-tree (right child). One or both can be NULL
• Each sub-tree is itself binary tree
• A null tree is a tree with no nodes
• Symmetry is not a tree requirement
24. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Height of Binary Tree:
• Considering number of nodes of binary tree as N,
• Maximum height Hmax is given as,
Hmax = N
• The tree with maximum height is rare, it occurs when
the entire tree is build in one direction
• Minimum height, Hmin is calculated as,
Hmin = [log2N] + 1
25. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Height of Binary Tree:
• Given a height of tree H,
• Minimum number of nodes in tree, Nmin = H
(in case of left skewed and right skewed binary tree)
• Maximum number of nodes in tree, Nmax = 𝟐𝑯 - 1
(in case of full binary tree)
• Example: Height of left hand side tree is 3, so minimum
number of nodes in tree = 3
• Height of right hand side tree is 3, so maximum number of
nodes in tree = 23-1 = 7
26. Quiz..
Data Structures Mrs. Kanchan Patil Department of Information Technology
• What will be the maximum and the minimum number of nodes in a binary tree of
height 5 are?
• Which of the following height is not possible for a binary tree with 50 nodes?
(A) 4 (B) 5 (C) 6 (D) None
Solution: Minimum height with 50 nodes = floor(log50) = 5. Therefore, height 4 is not
possible.
27. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Complete/Nearly Complete Binary Tree:
• A binary tree T is said to be complete binary tree if it has
maximum number of nodes for its height
• The maximum entry is reached when the last level is full
• A tree is considered nearly complete if it has minimum
height for it’s nodes and all the nodes in the last level are
found on left
• All the levels are completely filled except possibly the last
level and the last level has all keys as left as possible
Nearly Complete Binary Tree
Complete Binary Tree
28. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Full Binary Tree:
• A Binary Tree is a full binary tree if every node has 0 or 2 children
• It is a binary tree in which all nodes except leaf nodes have two
children
• Perfect Binary Tree:
• A Binary tree is a Perfect Binary Tree in which all the internal
nodes have two children and all leaf nodes are at the same level.
• It has 2i nodes at level i
• No. of leaf nodes = No. of internal nodes + 1
• Total no. of nodes = (2 * i) +1 where, i is internal node
Full Binary Tree
Perfect Binary Tree
29. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
31. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Balance Factor:
• The distance of a node from the root determines how efficiently it can be located
• The children of any node in a tree can be accessed by following only one path, the one
that needs to the desired node
• Shorter the tree, it is easier to locate any node in the tree
• To determine weather the tree is balanced, balance factor can be calculated
• Balance factor of a tree is the difference in height between its left and right sub-trees
• Balance factor, B can be determined by the formula,
B = HL – HR,
where, HL = Height of left subtree and HR is height of right subtree
32. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Balance Factor:
• The tree is said to be balanced, if its balance factor is 0 and its sub-trees are also
balanced.
• As, A binary tree is balanced if the height of its sub-trees differs by no more than one
(it can be -1, 0, +1) and its sub-trees are also balanced
A
33. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Example:
35. Properties of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Binary Tree Structure:
• Each node contains data and two pointers, one to point left sub-tree and another to
point right sub-tree
Node
• left subtree <pointer to node>
• data <data type>
• right subtree <pointer to node>
End node
36. Advantages of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Searching in Binary tree become faster
• Binary tree provides six traversals
• Two of six traversals give sorted order of elements
• Maximum and minimum elements can be directly picked up
• It is used for graph traversal and to convert an expression to postfix and prefix forms
37. Applications of Binary Tree
Data Structures Mrs. Kanchan Patil Department of Information Technology
• Search Engines: Organize and index web pages
• File System: Organize and index files
• Searching and Sorting: Binary Search, quicksort, merge sort
• Expression Tree: To represent mathematical expressions
• Huffman Coding: data compression
• Decision Tree: Machine Learning and Data Mining model
• Game AI: to model different possibilities and outcome of game
• TRIE: Storage and string retrieval
• Cryptography: generate and manage encryption, decryption keys
38. References
Data Structures Mrs. Kanchan Patil Department of Information Technology
• R. Lafore, “Data structures and Algorithms in Java”, Pearson education, ISBN: 9788
131718124.
• Michael Goodrich, Roberto Tamassia, Michael H. Goldwasser, “Data Structures and
Algorithms in Java”, 6th edition, wiley publication, ISBN: 978-1-118-77133-4
• R. Gilberg, B. Forouzan, “Data Structure: A Pseudo code approach with C++”, Cengage
Learning.
• Sartaj Sahni, “Data Structures, Algorithms and Applications in C++”, 2 nd Edition,
Universities Press.
• E. Horowitz, S. Sahni, S. Anderson-freed, “Fundamentals of Data Structures in C”, 2 nd
Edition, University Press, ISBN 978-81-7371-605-8.
• https://www.geeksforgeeks.org/difference-between-linear-and-non-linear-data-
structures/
• http://www.btechsmartclass.com/data_structures/tree-terminology.html
39. Data Structures Mrs. Kanchan Patil Department of Information Technology
Thank You!!!
Happy Learning!!!