This document summarizes a lecture on dynamic data structures and tree structures. It discusses different types of trees like binary trees and binary search trees. It describes how to traverse trees using preorder, inorder and postorder traversal. It also explains how to insert and delete nodes from a sorted binary tree in O(h(T)) time where h(T) is the height of the tree. Other tree types mentioned include 2-3 trees, AVL trees, B-trees and red-black trees.
This document discusses various dynamic data structures including pointers, linked lists, stacks, and queues. It provides details on pointers and how they allow dynamically allocated memory to be addressed. Linked lists, including singly linked, doubly linked, circular lists are covered. Implementation of stacks and queues as first-in last-out and first-in first-out data structures is also summarized. Examples of using each data structure are given.
In computer science, a data structure is a data organization, management, and storage format that enables efficient access and modification. More precisely, a data structure is a collection of data values, the relationships among them, and the functions or operations that can be applied to the data. https://apkleet.com
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This document introduces several common data structures used in computer science, including arrays, linked lists, stacks, queues, trees, and graphs. Arrays store a collection of elements of the same type in a linear order. Linked lists consist of nodes that contain data and links to other nodes, allowing efficient insertion and removal. Stacks and queues are linear data structures where elements can only be added or removed from one end, with stacks following last-in first-out order and queues following first-in first-out order. Trees store hierarchical relationships between elements, and graphs represent relationships between elements without a defined hierarchy.
The document discusses R-trees, a data structure used to index multi-dimensional spatial data. R-trees allow for efficient searching of spatial data by grouping data into minimum bounding rectangles (MBRs) and storing them in a tree structure based on these envelopes. The tree structure resembles a B+-tree, with internal nodes containing pointers to child nodes or data records. R-trees provide efficient search, insertion, and deletion of spatial data objects through operations on the tree structure and splitting or merging of nodes as needed.
The document discusses data structures and algorithms. It defines data structures as a means of storing and organizing data, and algorithms as step-by-step processes for performing operations on data. The document also discusses abstract data types which define the operations that can be performed on a data structure independently of its specific implementation. Common data structures like stacks, queues, and lists are classified and their algorithms and applications explained.
The document discusses data structures and their operations. It defines data structures as the logical organization of data and divides them into linear and non-linear structures. Linear structures include arrays, linked lists, stacks, and queues which store elements sequentially. Non-linear structures include trees and graphs which have indirect relationships between elements. Common operations on data structures include traversing, searching, insertion, deletion, sorting, and merging. The complexity of algorithms is also discussed in terms of time and space requirements. Elementary data organization involves grouping data into items, records, and files to structure information.
Introduction of Data Structures and Algorithms by GOWRU BHARATH KUMARBHARATH KUMAR
Basic Terminology: Elementary Data Organization:
Data Structures Usage
Data Structures Implementation
CLASSIFICATION OF DATA STRUCTURES
DATA STRUCTURES OPERATIONS
Space-Time Trade-off
Searching Algorithms
This document discusses various dynamic data structures including pointers, linked lists, stacks, and queues. It provides details on pointers and how they allow dynamically allocated memory to be addressed. Linked lists, including singly linked, doubly linked, circular lists are covered. Implementation of stacks and queues as first-in last-out and first-in first-out data structures is also summarized. Examples of using each data structure are given.
In computer science, a data structure is a data organization, management, and storage format that enables efficient access and modification. More precisely, a data structure is a collection of data values, the relationships among them, and the functions or operations that can be applied to the data. https://apkleet.com
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FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
This document introduces several common data structures used in computer science, including arrays, linked lists, stacks, queues, trees, and graphs. Arrays store a collection of elements of the same type in a linear order. Linked lists consist of nodes that contain data and links to other nodes, allowing efficient insertion and removal. Stacks and queues are linear data structures where elements can only be added or removed from one end, with stacks following last-in first-out order and queues following first-in first-out order. Trees store hierarchical relationships between elements, and graphs represent relationships between elements without a defined hierarchy.
The document discusses R-trees, a data structure used to index multi-dimensional spatial data. R-trees allow for efficient searching of spatial data by grouping data into minimum bounding rectangles (MBRs) and storing them in a tree structure based on these envelopes. The tree structure resembles a B+-tree, with internal nodes containing pointers to child nodes or data records. R-trees provide efficient search, insertion, and deletion of spatial data objects through operations on the tree structure and splitting or merging of nodes as needed.
The document discusses data structures and algorithms. It defines data structures as a means of storing and organizing data, and algorithms as step-by-step processes for performing operations on data. The document also discusses abstract data types which define the operations that can be performed on a data structure independently of its specific implementation. Common data structures like stacks, queues, and lists are classified and their algorithms and applications explained.
The document discusses data structures and their operations. It defines data structures as the logical organization of data and divides them into linear and non-linear structures. Linear structures include arrays, linked lists, stacks, and queues which store elements sequentially. Non-linear structures include trees and graphs which have indirect relationships between elements. Common operations on data structures include traversing, searching, insertion, deletion, sorting, and merging. The complexity of algorithms is also discussed in terms of time and space requirements. Elementary data organization involves grouping data into items, records, and files to structure information.
Introduction of Data Structures and Algorithms by GOWRU BHARATH KUMARBHARATH KUMAR
Basic Terminology: Elementary Data Organization:
Data Structures Usage
Data Structures Implementation
CLASSIFICATION OF DATA STRUCTURES
DATA STRUCTURES OPERATIONS
Space-Time Trade-off
Searching Algorithms
This document provides an overview and introduction to data structures. It discusses key terminology like data, data items, and fields. It also covers different types of data structures like linear (arrays, linked lists) and non-linear (trees, graphs) structures. Common data structure operations like traversing, searching, inserting and deleting are explained. The document stresses the importance of selecting the appropriate data structure based on the problem and required operations. It also briefly discusses algorithm design, implementation, testing, and analysis of time and space complexity.
Dynamic multi level indexing Using B-Trees And B+ TreesPooja Dixit
B-TREE, Properties of B-Tree, B-Tree of minimum degree 3, Drawbacks of B-Tree, B+ tree, B+ tree, Structure of the internal nodes of a B+ tree , structure of the leaf nodes of a B+ tree , Example of B+ tree
Introductiont To Aray,Tree,Stack, QueueGhaffar Khan
This document provides an introduction to data structures and algorithms. It defines key terminology related to data structures like entities, fields, records, files, and primary keys. It also describes common data structures like arrays, linked lists, stacks, queues, trees, and graphs. Finally, it discusses basic concepts in algorithms like control structures, complexity analysis, and examples of searching algorithms like linear search and binary search.
This document introduces data structures and their classifications. It defines data structure as a structured way of organizing data in a computer so it can be used efficiently. Data structures are classified as simple, linear, and non-linear. Linear structures like arrays, stacks, and queues store elements in a sequence while non-linear structures like trees and graphs have non-sequential relationships. The document discusses common operations on each type and provides examples of different data structures like linked lists, binary trees, and graphs. It concludes by noting data structures should be selected based on the nature of the data and requirements of operations.
This document discusses data structures and their applications. It defines key terms like data, data item, entity, attribute, field, record, and file. It explains that a data structure is a logical organization of data that specifies the data elements and operations that can be performed on them. Common operations include traversing, searching, inserting, and deleting. The choice of data structure depends on how frequently certain operations will be performed. Real-life data manipulation requires storage, retrieval, and transformation of user data.
Data Structure is the specific method for sorting out the data in a system with the goal that it could be utilized efficiently. These can implement at least one specific abstract data types (ADT), which indicate the operations that can be performed on the data structure and the computational unpredictability of those operations. Copy the link given below and paste it in new browser window to get more information on Data Structure & Algorithms:- www.transtutors.com/homework-help/computer-science/data-structure-and-algorithms.aspx
Furnish an Index Using the Works of Tree Structuresijceronline
This document discusses tree-based indexing schemes, specifically B-trees and B+-trees. It provides definitions and descriptions of the key components and properties of B-trees and B+-trees, including their nodes, keys, pointers, operations like search, insertion and deletion. Examples and figures are used to illustrate the concepts. The capacity and performance of B-trees and B+-trees are also analyzed and compared.
A data structure is a way of organizing data in a computer's memory so that it can be used efficiently by algorithms. The choice of data structure depends on the abstract data type and the operations that will be performed on the data. Some key characteristics of data structures include whether they are linear, static, homogeneous, or dynamic. Common operations on data structures include traversing, searching, inserting, deleting, sorting, and merging. The efficiency of sorting algorithms is analyzed based on best case, worst case, and average case time complexities, which typically range from O(n log n) to O(n2).
The document introduces different data structures commonly used in programming, including lists, arrays, stacks, queues, trees, and graphs. It provides examples of how queues and binary trees are structured and when each would be an appropriate data structure to use. Programmers must choose which data structure to implement based on the nature of the data and the necessary operations. Alice supports lists and arrays as basic built-in data structures that can be used to represent other data structures or organize program data.
This document provides an overview of a lecture on modern database systems. It discusses the following key points:
1) The lecture will review relational models, SQL, storage and indexing. Assignments will be posted online and are due on February 12th.
2) The relational model represents data in tables and supports intuitive querying with SQL. Common queries like selections, projections, joins and aggregations are demonstrated.
3) Database files can be organized using heap files, sorted files and indexes like B+ trees and hash indexes. These different structures allow for efficient retrieval of data based on queries.
1. The document defines and provides examples of various data structures including lists, stacks, queues, trees, and their properties.
2. Key concepts covered include linear and non-linear data structures, common tree types, tree traversals, and operations on different data structures like insertion, deletion, and searching.
3. Examples are provided to illustrate concepts like binary search trees, tree representation and traversal methods.
Data structure,abstraction,abstract data type,static and dynamic,time and spa...Hassan Ahmed
The document summarizes a group project submitted by 5 students on basic data structures. It discusses topics like stacks, queues, linked lists, and the differences between static and dynamic data structures. It provides examples and definitions of basic linear data structures like stacks, queues, and deques. It also explains how insertions and removals work differently in static versus dynamic data structures due to their fixed versus flexible memory allocation.
This document discusses data structures and their applications. It defines objects, classes, inheritance, and interfaces. It discusses the major data structures used in relational database management systems, network data models, and hierarchical data models. It also discusses linked lists, stacks, queues, trees and graphs. It provides examples of linear and non-linear data structures as well as static and dynamic data structures.
This document provides an introduction to data structures and discusses various concepts related to data structures including:
1) Classifications of data structures as primitive and non-primitive, linear and non-linear.
2) Common operations on data structures like traversing, searching, inserting, deleting etc.
3) Pointers and dynamic memory allocation in C using functions like malloc(), calloc(), realloc() and free().
The document discusses height balanced binary trees. A height balanced binary tree is one where, for each node, the heights of the left and right subtrees differ by no more than 1. An AVL tree is a type of height balanced binary tree. The document provides an example of a height balanced tree that is not completely balanced. It also gives a formula to calculate the maximum number of nodes in a balanced binary tree of height h as 2h-1 - 1 internal nodes plus up to 2h leaf nodes.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
The document describes a proposed path indexing technique for efficiently processing path queries in graph databases. The path index would store paths of length up to K from the graph. Given a pattern of a path, the index would return all graph paths matching that pattern faster than querying the graph database directly. The path index represents paths as keys consisting of the path ID and node IDs. These keys are stored in a B+tree structure for fast retrieval of matching paths when given a query pattern. Compression techniques can reduce the size of keys since the node IDs are often sequential.
This document provides an introduction to data structures. It defines data structures as a way of organizing data so that it can be used efficiently. The document then discusses basic terminology, why data structures are important, how they are studied, and how they are classified as simple or compound, and linear or non-linear. It proceeds to describe common data structures like arrays, stacks, queues, linked lists, trees, and graphs, and how they support basic operations. The document concludes by discussing how to select an appropriate data structure based on the problem constraints and required operations.
1) The document is an assignment submission on data structures covering queues, linked lists, and trees.
2) It discusses the concepts of queues including FIFO operations, applications such as call centers, and implementations using arrays or linked lists.
3) Linked lists are described as dynamic structures that connect nodes through pointers and are used for stacks, queues, and graphs. Common types include singly, doubly, and circular linked lists.
4) Trees are hierarchical structures where each node has a maximum of two children. Binary trees, their terminology, types such as full and balanced binary trees, and advantages are outlined.
This document discusses trees and binary trees. It defines key terminology like root, leaf nodes, internal nodes, and provides examples of different types of binary trees including full, complete, balanced, and binary search trees. It also covers different representations of binary trees using arrays and linked lists as well as common traversal techniques for binary trees like preorder, inorder and postorder traversal.
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The document discusses binary search trees, which are binary trees that store keys in nodes such that all keys in the left subtree of a node are less than the node's key and all keys in the right subtree are greater. This property allows searching, insertion, and deletion operations to be performed efficiently in O(log N) time on average. The document covers tree terminology, traversal methods, and implementations of searching, inserting, deleting, finding the minimum/maximum keys, and maintaining the binary search tree properties during operations.
This document provides an overview and introduction to data structures. It discusses key terminology like data, data items, and fields. It also covers different types of data structures like linear (arrays, linked lists) and non-linear (trees, graphs) structures. Common data structure operations like traversing, searching, inserting and deleting are explained. The document stresses the importance of selecting the appropriate data structure based on the problem and required operations. It also briefly discusses algorithm design, implementation, testing, and analysis of time and space complexity.
Dynamic multi level indexing Using B-Trees And B+ TreesPooja Dixit
B-TREE, Properties of B-Tree, B-Tree of minimum degree 3, Drawbacks of B-Tree, B+ tree, B+ tree, Structure of the internal nodes of a B+ tree , structure of the leaf nodes of a B+ tree , Example of B+ tree
Introductiont To Aray,Tree,Stack, QueueGhaffar Khan
This document provides an introduction to data structures and algorithms. It defines key terminology related to data structures like entities, fields, records, files, and primary keys. It also describes common data structures like arrays, linked lists, stacks, queues, trees, and graphs. Finally, it discusses basic concepts in algorithms like control structures, complexity analysis, and examples of searching algorithms like linear search and binary search.
This document introduces data structures and their classifications. It defines data structure as a structured way of organizing data in a computer so it can be used efficiently. Data structures are classified as simple, linear, and non-linear. Linear structures like arrays, stacks, and queues store elements in a sequence while non-linear structures like trees and graphs have non-sequential relationships. The document discusses common operations on each type and provides examples of different data structures like linked lists, binary trees, and graphs. It concludes by noting data structures should be selected based on the nature of the data and requirements of operations.
This document discusses data structures and their applications. It defines key terms like data, data item, entity, attribute, field, record, and file. It explains that a data structure is a logical organization of data that specifies the data elements and operations that can be performed on them. Common operations include traversing, searching, inserting, and deleting. The choice of data structure depends on how frequently certain operations will be performed. Real-life data manipulation requires storage, retrieval, and transformation of user data.
Data Structure is the specific method for sorting out the data in a system with the goal that it could be utilized efficiently. These can implement at least one specific abstract data types (ADT), which indicate the operations that can be performed on the data structure and the computational unpredictability of those operations. Copy the link given below and paste it in new browser window to get more information on Data Structure & Algorithms:- www.transtutors.com/homework-help/computer-science/data-structure-and-algorithms.aspx
Furnish an Index Using the Works of Tree Structuresijceronline
This document discusses tree-based indexing schemes, specifically B-trees and B+-trees. It provides definitions and descriptions of the key components and properties of B-trees and B+-trees, including their nodes, keys, pointers, operations like search, insertion and deletion. Examples and figures are used to illustrate the concepts. The capacity and performance of B-trees and B+-trees are also analyzed and compared.
A data structure is a way of organizing data in a computer's memory so that it can be used efficiently by algorithms. The choice of data structure depends on the abstract data type and the operations that will be performed on the data. Some key characteristics of data structures include whether they are linear, static, homogeneous, or dynamic. Common operations on data structures include traversing, searching, inserting, deleting, sorting, and merging. The efficiency of sorting algorithms is analyzed based on best case, worst case, and average case time complexities, which typically range from O(n log n) to O(n2).
The document introduces different data structures commonly used in programming, including lists, arrays, stacks, queues, trees, and graphs. It provides examples of how queues and binary trees are structured and when each would be an appropriate data structure to use. Programmers must choose which data structure to implement based on the nature of the data and the necessary operations. Alice supports lists and arrays as basic built-in data structures that can be used to represent other data structures or organize program data.
This document provides an overview of a lecture on modern database systems. It discusses the following key points:
1) The lecture will review relational models, SQL, storage and indexing. Assignments will be posted online and are due on February 12th.
2) The relational model represents data in tables and supports intuitive querying with SQL. Common queries like selections, projections, joins and aggregations are demonstrated.
3) Database files can be organized using heap files, sorted files and indexes like B+ trees and hash indexes. These different structures allow for efficient retrieval of data based on queries.
1. The document defines and provides examples of various data structures including lists, stacks, queues, trees, and their properties.
2. Key concepts covered include linear and non-linear data structures, common tree types, tree traversals, and operations on different data structures like insertion, deletion, and searching.
3. Examples are provided to illustrate concepts like binary search trees, tree representation and traversal methods.
Data structure,abstraction,abstract data type,static and dynamic,time and spa...Hassan Ahmed
The document summarizes a group project submitted by 5 students on basic data structures. It discusses topics like stacks, queues, linked lists, and the differences between static and dynamic data structures. It provides examples and definitions of basic linear data structures like stacks, queues, and deques. It also explains how insertions and removals work differently in static versus dynamic data structures due to their fixed versus flexible memory allocation.
This document discusses data structures and their applications. It defines objects, classes, inheritance, and interfaces. It discusses the major data structures used in relational database management systems, network data models, and hierarchical data models. It also discusses linked lists, stacks, queues, trees and graphs. It provides examples of linear and non-linear data structures as well as static and dynamic data structures.
This document provides an introduction to data structures and discusses various concepts related to data structures including:
1) Classifications of data structures as primitive and non-primitive, linear and non-linear.
2) Common operations on data structures like traversing, searching, inserting, deleting etc.
3) Pointers and dynamic memory allocation in C using functions like malloc(), calloc(), realloc() and free().
The document discusses height balanced binary trees. A height balanced binary tree is one where, for each node, the heights of the left and right subtrees differ by no more than 1. An AVL tree is a type of height balanced binary tree. The document provides an example of a height balanced tree that is not completely balanced. It also gives a formula to calculate the maximum number of nodes in a balanced binary tree of height h as 2h-1 - 1 internal nodes plus up to 2h leaf nodes.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
The document describes a proposed path indexing technique for efficiently processing path queries in graph databases. The path index would store paths of length up to K from the graph. Given a pattern of a path, the index would return all graph paths matching that pattern faster than querying the graph database directly. The path index represents paths as keys consisting of the path ID and node IDs. These keys are stored in a B+tree structure for fast retrieval of matching paths when given a query pattern. Compression techniques can reduce the size of keys since the node IDs are often sequential.
This document provides an introduction to data structures. It defines data structures as a way of organizing data so that it can be used efficiently. The document then discusses basic terminology, why data structures are important, how they are studied, and how they are classified as simple or compound, and linear or non-linear. It proceeds to describe common data structures like arrays, stacks, queues, linked lists, trees, and graphs, and how they support basic operations. The document concludes by discussing how to select an appropriate data structure based on the problem constraints and required operations.
1) The document is an assignment submission on data structures covering queues, linked lists, and trees.
2) It discusses the concepts of queues including FIFO operations, applications such as call centers, and implementations using arrays or linked lists.
3) Linked lists are described as dynamic structures that connect nodes through pointers and are used for stacks, queues, and graphs. Common types include singly, doubly, and circular linked lists.
4) Trees are hierarchical structures where each node has a maximum of two children. Binary trees, their terminology, types such as full and balanced binary trees, and advantages are outlined.
This document discusses trees and binary trees. It defines key terminology like root, leaf nodes, internal nodes, and provides examples of different types of binary trees including full, complete, balanced, and binary search trees. It also covers different representations of binary trees using arrays and linked lists as well as common traversal techniques for binary trees like preorder, inorder and postorder traversal.
mitochondria moment and super computer integration.pptAMMAD45
The document discusses binary search trees, which are binary trees that store keys in nodes such that all keys in the left subtree of a node are less than the node's key and all keys in the right subtree are greater. This property allows searching, insertion, and deletion operations to be performed efficiently in O(log N) time on average. The document covers tree terminology, traversal methods, and implementations of searching, inserting, deleting, finding the minimum/maximum keys, and maintaining the binary search tree properties during operations.
This document discusses binary search trees. It begins by explaining what trees and binary trees are. It then describes key properties of binary search trees like their height and how every node's key is greater than all keys in its left subtree and less than keys in its right subtree. The document outlines common binary search tree operations like searching, insertion, deletion and traversal. It explains how each operation takes O(height of the tree) time since they involve traversing down a path in the tree from the root.
Binary trees and binary search trees are discussed. Binary trees have nodes with at most two children, while binary search trees have the additional property that for every node, all keys in its left subtree are smaller than the node's key and all keys in its right subtree are larger. Common tree operations like searching, insertion, and deletion can be performed in O(log n) time on balanced binary search trees. AVL trees are discussed as one way to balance binary search trees through rotations.
non linear data structure -introduction of treeSiddhi Viradiya
The document defines trees and binary trees. A tree consists of nodes connected by branches, with one root node and zero or more subtrees. A binary tree restricts each node to have at most two children. The document discusses tree terminology like root, child, parent, leaf nodes. It also covers tree traversal methods like preorder, inorder and postorder. Finally, it shows how to construct a binary tree from its traversals.
The document discusses various tree data structures including binary trees and their terminology. It defines a tree as a set of nodes connected by links/branches where one node is designated as the root. Key terms discussed include child, parent, leaf, root, and level. The document also covers different ways to represent trees using arrays and linked lists and how to traverse trees using preorder, inorder, and postorder traversal algorithms.
Binary trees are a non-linear data structure that impose a hierarchical structure on a collection of items. They consist of nodes connected by edges, with one root node and no cycles. Binary trees have many applications including representing directory structures, organizational charts, and mathematical expressions. Common tree terminology includes root, child, parent, leaf, internal node, and sibling nodes. Traversing a binary tree involves visiting each node exactly once using preorder, inorder, or postorder traversal methods.
The document discusses various tree data structures and their terminology. It begins by defining a tree as a set of nodes connected in a parent-child relationship, with one root node and multiple disjoint subtree structures. Key terminology introduced includes root, child, parent, leaf nodes, siblings, ancestors, and height/depth. Binary trees are defined as having at most two children per node. Common traversal orders of trees - preorder, inorder, and postorder - are explained along with examples. Finally, algorithms for traversing binary trees using stacks are presented for preorder and inorder traversal.
Binary search trees have the following key properties:
1. Each node contains a value.
2. The left subtree of a node contains only values smaller than the node's value.
3. The right subtree of a node contains only values larger than the node's value.
Binary search trees allow for efficient insertion and search operations in O(log n) time due to their structure. Deletion may require rebalancing the tree to maintain the binary search tree properties.
This document defines and provides examples of trees and binary trees. It begins by defining trees as hierarchical data structures with nodes and edges. It then provides definitions for terms like path, forest, ordered tree, height, and multiway tree. It specifically defines binary trees as having two children per node. The document gives examples and properties of binary trees, including full, complete, and binary search trees. It also explains linear and linked representations of binary trees and different traversal methods like preorder, postorder and inorder. Finally, it provides examples of insertion and deletion operations in binary search trees.
The document discusses different types of tree data structures, focusing on binary trees. It defines a binary tree recursively as a finite set of elements that is either empty or partitioned into three disjoint subsets containing a single root element and left and right subtrees. The document outlines common binary tree terminology like nodes, parents, descendants, and levels. It also describes complete binary trees where all levels are fully filled except the lowest, which has nodes filled from left to right.
The document discusses binary tree traversal methods. It defines key binary tree terminology like nodes, edges, root, and provides examples of different types of binary trees like strictly binary, complete, and almost complete binary trees. It also explains the three common traversal techniques for binary search trees - in-order, pre-order and post-order traversals - and provides pseudocode algorithms and examples for each traversal method.
The document discusses binary trees and binary search trees. It begins with definitions of tree, binary tree, and binary search tree. It describes the key properties and terminology used for trees including nodes, degrees, heights, paths, etc. It then covers various tree traversal methods like preorder, inorder and postorder traversal. Operations for binary search trees like searching, insertion and deletion of nodes are explained along with algorithms. Different representations of binary trees using arrays and linked lists are also presented.
A binary search tree is a binary tree where each node has at most two children, and the value of each node is greater than all values in its left subtree and less than all values in its right subtree. This structure allows for efficient search, insertion, and deletion operations in O(log n) time due to the ordering of nodes. Common operations on binary search trees include searching for a node, counting the number of nodes, inserting a new node, and deleting a node while maintaining the binary search tree properties.
Tree and Binary search tree in data structure.
The complete explanation of working of trees and Binary Search Tree is given. It is discussed such a way that everyone can easily understand it. Trees have great role in the data structures.
The document discusses various types of tree data structures, including binary trees and binary search trees. It provides examples of how to implement binary search trees using a Node class, including how to search, insert, and delete nodes. It also discusses tree traversal algorithms and other applications of trees such as parsing, XML representation, and graphs.
The document defines and explains key concepts related to binary trees including their terminology, properties, and different types. It discusses binary tree concepts such as parent, child, root, leaf nodes, subtrees, and full and complete binary trees. The document also covers spanning trees, minimal spanning trees, Prim's and Kruskal's algorithms for finding minimum spanning trees, and binary search trees.
This document discusses trees as a data structure. It defines trees as structures containing nodes where each node can have zero or more children and at most one parent. Binary trees are defined as trees where each node has at most two children. The document discusses tree terminology like root, leaf, and height. It also covers binary search trees and their properties for storing and searching data, as well as algorithms for inserting and deleting nodes. Finally, it briefly discusses other types of trees like balanced search trees and parse trees.
This document provides information about tree data structures. It defines key tree terminology like root, parent, child, leaf node, internal node, subtree, and siblings. It also describes different types of trees like binary trees, binary search trees, threaded binary trees, and heaps. Common tree traversal algorithms like preorder, inorder, and postorder are explained. Priority queues and their representations using heaps are also discussed.
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Lecture4b dynamic data_structure
1. Lecture 4
Dynamic Data Structure Part 2
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 1
2. Content Lecture 4 Dynamic Data
Part 2
Tree structures
General
Tree Representation
Balanced Trees
Binary Tree
Sorted Binary Tree
Binary Search Tree
Insert an element in a Sorted Binary Tree
Delete a node in a Sorted Binary Tree
Other Tree Types
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 2
3. Tree structures
General
Trees are one of the most important data structures
There are several different variants
We will have a closer look to some of them
Trees are used in computer science for data storage,
searching and sorting
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 3
4. Tree structures
Definition
A Tree is a finite amount T of one or several nodes such that
There is a significant node called root(T)
The other nodes can be divided in disjunctive amount
T1, .. Tm
Each of these amounts is again a Tree
These Trees are called Sub Trees
This is a recursive definition
At the end a Tree with only one node remains
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 4
5. Tree Representation
You can display Trees in different ways
In this course outline Trees are illustrated with the root on
the top and the Sub Trees below
This is a common illustration
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 5
FED
B C
A
6. Tree Representation
These are some different representations
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 6
7. Tree Representation
Family Trees
A common example for trees are Family Trees
Two variants of Family Trees exist:
Starting from an ancestor as root and illustrated all the
descendants
Starting from a descendant as root and illustrated all the
ancestors
In Family Trees you can have redundancies if some of the
ancestors have the same ancestors
In this case the entries represent the role of the ancestor
(for example grandmother on the mother’s side)
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 7
10. Tree structures
Definitions for Trees
Each node in a Tree is the root of one of the Sub Trees
Each Tree has zero or more child nodes which are below in
the Tree
A node with a child node is called parent node to the child
(also ancestor node or superior)
A root node is a node with no parents
Each Tree has at least one root node
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 10
12. Tree structures
The depth of a node is the length of the path to its root
A node p is an ancestor to node q if p exists on the path
from q to root
The node q is called a descendant of p
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 12
14. Tree structures
The size of a node is the number of descendants a node has
including itself
Siblings are nodes that share the same parent node
Nodes without a child node are called leaf node or
terminal nodes
Internal or inner node are nodes with one or more child
nodes therefore size > 0
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 14
16. Tree structures
In-degree of a node is the number of edges arriving at that
node
The only node with In-degree = 0 is the root node
Out-degree of a node is the number of edges leaving that
node
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 16
18. Tree structures
The level of a node is defined
as:
The level of the root(T) is 0
The level of any other node
is increased by 1 to the
level of the node which is
the root of the superior sub
tree
level(node) = level(parent) + 1
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 18
G
FED
B C
A Level 0
Level 1
Level 2
Level 3
19. Balanced Trees
If the relative order of the Sub Trees is important you call it
a Balanced Tree
If the order is not important this is called an Oriented
Tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 19
G
FED
B C
A
G
EFD
B C
A
These Trees are
different if you look at
them as Balanced Tree
but they are consider
as the same if you look
at them as Oriented
Trees
20. Binary Trees
A Binary Tree is a finite amount of nodes where the
amount is empty or contains a root and two disjunctive
Binary Trees
These two Binary Trees are called left and right Sub Tree
of the root
That means that every node of the tree has zero, one or two
child nodes
Binary Trees are not a special case of Trees in general
For example you can have an empty amount as a Binary
Tree but not as a General Tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 20
21. Binary Trees
Any data in the Tree structure can be reached by starting at
the rood node and following either the left or the right
child
Binary Trees are used for implementation of Binary Tree
Search and Binary Heaps
Examples for Binary Trees are the elimination contest in
tennis or other sport competitions
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 21
22. Binary Trees
Example
These two Binary Trees are not the same
One has left and the other a right Sub Tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 22
B
A
B
A
Binary tree with a
left sub tree
Binary tree with a
right sub tree
23. Binary Trees
Implementation
Binary Trees are easily implemented as a data type like
One pointer points to the root
If this pointer is null than the Tree is empty
A node in the Binary Tree contains a pointer to an object
or a record and the two pointers to the left and the right
Sub Tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 23
25. Binary Trees
There are three possibilities to traverse a Binary Tree:
preorder, inorder and postorder
If the Tree is empty there is nothing to do
Otherwise the traverse possibilities are defined as:
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 25
preorder visit the root
traverse the left sub tree
traverse the right sub tree
inorder traverse the left sub tree
visit the root
traverse the right sub tree
postorder traverse the left sub tree
traverse the right sub tree
visit the root
B C
A
A B C
B A C
B C A
26. Binary Trees
Examples
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 26
preorder: A B D C E G F H I
inorder: D B A E G C H F I
postorder: D B G E H I F C A
G
FED
B C
A
IH
Note
The preorder traverse is also called Polish
notation (after the Polish logician Jan
Lukasiewicz) and the postorder traverse is
therefore called a backwards Polish
notation
28. Sorted Binary Tree
A tree T is a Sorted Binary Tree where
v: T V function to receive the value of a node
T a Binary Tree
T1 the left Sub Tree (also a Sorted Binary Trees)
T2 the right Sub Tree (also a Sorted Binary Trees) such that:
v(root(T)) > v(t) for all nodes t in T1
v(root(T)) ≤ v(t) for all nodes t in T2
If you traverse a Sorted Binary Tree inorder, than all values
are reached in a sorted way
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 28
30. Binary Tree
Definitions
The height of a Tree is the length of the path from the root to
the deepest node in the Tree:
h(T) = 0 if the Tree T is empty
h(T) = max(h(T1), h(T2)) + 1
where T1 is the left and T2 the right Sub Tree
The cardinality of a Tree T is the number n of elements in
the Tree that means:
n = card(T)
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 30
31. Binary Tree
Example
The height of this tree is 4
The cardinality of this tree is 7
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 31
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32. Binary Tree
A Tree is called height-balancing if the Tree T is empty or if
| h(T1) – h (T2) | ≤ 1
where h is the height of a Tree and also the Sub Trees T1 and
T2 are height-balancing Trees
A Tree is called balanced if the Tree T is empty or if
| card(T1) – card(T2) | ≤ 1
and also the Sub Trees T1 and T2 are balanced
If a Tree T is balance he is also balanced in its height
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 32
33. Sorted Binary Tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 33
height-balancing Tree but not a balanced
height left tree = 2, height right tree = 3
|h(T1) – h(T2) = |2 – 3| = 1
card left tree = 2, card right tree = 4
|card(T1) – card(T2) = |2 – 4| = 2
Balanced Tree ( height-balancing)
height left tree = 2, height right tree = 2
|h(T1) – h(T2) = |2 – 2| = 0
card left tree = 3, card right tree = 3
|card(T1) – card(T2) = |3 – 3| = 0
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5
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2 6
4
3
34. Binary Search Tree
k a value and the
s search element
v(s) = k
T (Binary Tree) with t = root(T)
If T is empty than there is nothing to do
If v(t) = k than the search element is found
If not than
Replace T with the left subtree T1 if k < v(t) T = T1
Replace T with the right subtree T2 if k ≥ v(t) T = T2
The effort for the search is O(h(T))
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 34
35. Binary Search Tree
Example
Search element k
k = 1
t = 5 1 < 5
t = 3 1 < 3
t = 1 1 = 1 element found!
k = 7
t = 5 7 > 5
t = 8 7 < 8
t = 6 7 > 6
t = 7 7 = 7 element found!
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 35
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36. Insert a element in a Sorted
Binary Tree
u a values with k = v(u) to be insert
T a tree with t the root of Tree T
Create a new binary sorted tree U containing only the root
u
If T is empty replace T by U
Otherwise
replace T with the left sub tree T1 if k < v(t) T = T1
replace T with the right sub tree T2 if k ≥ v(t) T = T2
The effort is also O(h(T))
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 36
37. Insert a element in a Sorted
Binary Tree
Example
Insert in an empty Tree the elements 5, 1, 3, 2, 8, 4, 6, 7 and 9:
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 37
5
1
3
2 4
9
8
7
6
38. Delete a node in a Sorted Binary
Tree
Be u ϵ T the node to be removed from Tree T
Be U the sub tree from T with root(U) = u
If the left Sub Tree U1 of U is empty than replace U by U1
If the right Sub Tree U2 of U is empty than replace U by U2
If both Sub Trees are not empty than:
Chose an u2 ϵ U2 such that v(u2) ≤ v(t) for all t ϵ U2
u2 has to be removed from U2
u2 replace the node u
The effort to delete a node in a Tree is O(h(T)) due to the
search for the minimal element u2
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 38
39. Delete a node in a Sorted Binary
Tree
Example
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 39
40. Other Tree Types
2-3 tree
2-3-4 tree
AA tree
AVL tree
B-tree
Elastic binary tree
Random binary tree
Red-black tree
Self-balancing binary search tree
Unrooted binary tree
23/10/2018 Lecture 4 Dynamic Data Structure Part 2 40