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La Teoría de Grafos es el lenguaje para describir y operar con redes de puntos y enlaces.

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Graph Theory Defined and Applications | Built In

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Graph theory is the study of graph data structures, which model the relationships between objects using vertices (nodes) connected by edges. It is a helpful tool to quantify and simplify complex systems.

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Aborda cómo almacenar eficientemente las conexiones de un grafo, comparando opciones como matrices y listas de adyacencia por su impacto en memoria y velocidad.

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5.2 REPRESENTACIÓN DE LOS GRAFOS. Los grafos se pueden representar de diferentes maneras, como por ejemplo: Representación por inci…

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Explora cómo recorrer grafos, identificar sus conexiones y sus subestructuras usando algoritmos clave como DFS y BFS.

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En Teoría de grafos , un Grafo ciclo o simplemente ciclo es un grafo que se asemeja a un polígono de n lados. Consiste en un camin…

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Grafos Eulerianos y Hamiltonianos exploran la fascinante búsqueda de un recorrido único, ya sea pasando por cada arista o visitando cada nodo, sin repetición.

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Ruiz Murcia, Antonio ; Aranda Ballesteros, F. Damián

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4. Grafos eulerianos y hamiltonianos | MG1 - Situación de aprendizaje 2.5: Relaciones y medida: Grafos en marcha para la movilidad urbana

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El grado de un vértice o nodo en un grafo es el número de conexiones directas que tiene un vértice con otros vértices en el grafo. El grado de un vértice se puede considerar como una medida de su conectividad dentro del grafo.

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Son estructuras jerárquicas de datos que organizan nodos (desde la raíz hasta las hojas). Su estudio abarca sus diversas formas y los métodos clave (preorden, inorden, postorden) para explorarlos sistemáticamente.

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Son estructuras de árbol avanzadas que garantizan balanceo y eficiencia en la gestión optimizada de datos, esenciales para bases de datos.

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La Teoría de Grafos es la clave para entender, modelar y optimizar las conexiones y redes que definen nuestro mundo.

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Introducción a Grafos - Formas de representación

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La conectividad se refiere a la propiedad de un grafo de tener sus elementos unidos. Un grafo conexo es aquel donde todos sus nodos están interconectados, permitiendo alcanzar cualquier punto desde cualquier otro. Si el grafo no es completamente conexo, las componentes conexas son sus subgrupos de nodos que sí están interconectados internamente, pero aislados del resto.

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In Mathematics, the meaning of connectivity is one of the fundamental concepts of graph theory. It demands a minimum number of elements (nodes or edges) that require to be removed to isolate the remaining nodes into separated subgraphs. It is closely related to the principles of network flow problems. The connectivity of a graph is an essential measure of its flexibility as a network.

The word " connectivity " may belong to several applications in day to day life. Usually, it is referred to as the connection between two or more things or properties. In terms of different subjects, the definition of connectivity is described below:

1. In topology, the connected space is the topological space, which we cannot write in the form of union of two or more open non-empty subsets.

2. In the field of information and technology, the connectivity may relate to internet connectivity by which several individual computers, cell phones and LANs are connected to the global Internet.

3. In the field of transport planning, connectivity is also called permeability which relates to the limit to which urban structures limit the movement of vehicles and people.

4. The concept of pixel connectivity is used in the field of image processing.

5. Most importantly, in Mathematics, the term connectivity is utilized in graph theory. Graph connectivity is applicable in routing, network, network tolerance, transportation network, etc.

Connectivity Definition

Connectivity is one of the essential concepts in graph theory. A graph may be related to either connected or disconnected in terms of topological space. If there exists a path from one point in a graph to another point in the same graph, then it is called a connected graph. Else, it is called a disconnected graph. Below are the diagrams which show various types of connectivity in the graphs.

Connectivity in Graph Theory

A graph is a connected graph if, for each pair of vertices, there exists at least one single path which joins them. A connected graph may demand a minimum number of edges or vertices which are required to be removed to separate the other vertices from one another. The graph connectivity is the measure of the robustness of the graph as a network.

In a connected graph, if any of the vertices are removed, the graph gets disconnected. Then the graph is called a vertex-connected graph. On the other hand, when an edge is removed, the graph becomes disconnected. It is known as an edge-connected graph.

Also, read: Types Of Graphs

Properties of Connectivity

The connected graph is called an undirected graph, which has at least one path between each pair of vertices.

The graph that is connected by three vertices is called 1-vertex connected graph since the removal of any of the vertices will lead to disconnection of the graph.

If in a connected graph, the removal of one edge leads to the disconnection of the graph, such a graph is called 1-edge connected graph.

If there exists a set (say S) of edges (or vertices) in a connected graph, such that by removing all the edges of set S will result in a disconnected graph. Then the set S is called a cut set. If S consists of vertices, then it is called a vertex-cut set. Similarly, if it has edges, then it is called an edge-cut set.

A bi-connected graph is a connected graph which has two vertices for which there are two disjoint paths between these two vertices.

Connected Graphs in Discrete Maths

There are different types of connected graphs explained in Maths. They are:

Fully Connected Graph

K-connected Graph

Strongly Connected Graph

Let us learn them one by one.

Fully Connected Graph

In graph theory, the concept of a fully-connected graph is crucial. It is also termed as a complete graph. It is a connected graph where a unique edge connects each pair of vertices. In other words, for every two vertices of a whole or a fully connected graph, there is a distinct edge.

A fully connected graph is denoted by the symbol K n, named after the great mathematician Kazimierz Kuratowski due to his contribution to graph theory. A complete graph K n possesses n/2(n−1) number of edges. Given below is a fully-connected or a complete graph containing 7 edges and is denoted by K ₇.

K connected Graph

A graph is called a k-connected graph if it has the smallest set of k-vertices in such a way that if the set is removed, then the graph gets disconnected. Complete or fully-connected graphs do not come under this category because they don't get disconnected by removing any vertices. A set of graphs has a large number of k vertices based on which the graph is called k-vertex connected. It could be one-connected, two-connected or bi-connected, three-connected or tri-connected.

Strongly Connected Graph

A graph can be defined as a strongly connected graph if its every vertex can be reached from every other vertex in the graph. In other words, any directed graph is called strongly connected if there exists a path in each possible direction between each pair of vertices in the graph.

In a graph (say G) which may not be strongly connected itself, there may be a pair of vertices say (a and b) that are called strongly connected to each other if in case there exists a path in all the possible directions between a and b.

Examples of Connectivity
Q.1: If a complete graph has a total of 20 vertices, then find the number of edges it may contain.
Solution: The formula for the total number of edges in a k ₁₅ graph is given by;

Number of edges = n(n-1)/2

= 20(20-1)/2

=10(19)

=190

Hence, it contains 190 edges.
Q.2: If a graph has 40 edges, then how many vertices does it have?
Solution: As we know,

Number of edges = n (n-1)/2

40 = n(n-1)/2

n(n-1) = 80

On solving the above quadratic equation, we get;

n ≈ 9.45, -8.45

Since, the number of vertices cannot be negative.

Therefore, n ≈ 9

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In Mathematics, the meaning of connectivity is one of the fundamental concepts of graph theory. It demands a minimum number of elements (nodes or edges) that require to be removed to isolate the remaining nodes into separated subgraphs.

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En un grafo, un camino es una secuencia de nodos conectados por aristas. Un camino simple es un camino donde ningún nodo se repite. Un ciclo es un camino que comienza y termina en el mismo nodo, formando un bucle cerrado.

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Búsqueda en grafos # Ahora, sabemos que podemos modelar problemas con grafos y cómo guardarlos de forma eficiente pero no podemos hacer nada aún con ellos. Estos primeros algoritmos que veremos ahora son esenciales y se usan con demasiada frecuencia así que es importante entenderlos bien.
Para estos dos algoritmos (y en general) usaremos listas de adyacencia, dado que de esta manera es mucho más eficiente listar los vecinos de un nodo.

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El Algoritmo de Fleury es un método para encontrar un recorrido Euleriano (que visita cada arista una sola vez) en un grafo. Su regla fundamental es evitar usar aristas "puente" (que desconectan el grafo) a menos que no haya otra opción. Solo aplica si el grafo es realmente Euleriano.

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Euler Circuits and Paths: Fleury’s Algorithm | Baeldung on Computer Science

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Explore the Fleury’s algorithm and its real-world applications.

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Árboles en Matemáticas Discretas: Teoría, Tipos | StudySmarter

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Árboles en Matemáticas Discretas: ✓ ‘Grafos’ ✓ ‘Teoría de Grafos’ ✓ ‘Combinatoria’ ✓ ‘Árboles Binarios’ StudySmarterOriginal!

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Un Árbol Binario de Búsqueda (ABB) es un árbol binario (cada nodo tiene hasta dos "hijos") donde los valores están ordenados: el valor de cada nodo es mayor que todos los de su subárbol izquierdo y menor que todos los de su subárbol derecho. Esto permite buscar, insertar y eliminar elementos de forma muy eficiente.

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In this free online class, BYJU’S Exam Prep GATE expert DV Sridhar Sir will Discuss the “Binary Tree Traversals Concept” in Data Structure & Algorithms for the GATE Computer Science Engineering (CSE) 2023 Exam Aspirants. BYJU’S Exam Prep GATE (Formerly Gradeup GATE) brings you the “Mastering Concept” series for💥GATE 2023/24💥and has launched new batches of FREE LIVE/Online classes for GATE 2023 to ensure sound preparation for the aspirants from the comfort of their homes. This is a dynamic course where students can watch LIVE YouTube classes delivered by experts. Additional to this, students will also get an opportunity to questions practice & doubts solving class. The session will be delivered by our experienced faculties of BYJU’S Exam Prep GATE (Formerly Gradeup GATE), who have vast teaching experience in the domain of GATE and other competitive examinations. Join our Free online LIVE session for more information regarding the GATE 2023 exam preparation strategy, the right time to start, tips and tricks, and share this session with your friends and colleagues, so that they can also get this important guidance from our experts do not forget to hit the like and subscribe button for more information and exam preparation. 📚 👉🏼 Meet our GATE 2022 Toppers | Topper Talk’s |BYJU’S GATE📚 👉https://bit.ly/3yMCf6I ***************************************************************** Are you preparing for GATE 2023, Join our LIVE session and share with your friends and colleagues so that they can also get this important guidance for GATE exam 2023 from our experts, and do not forget to hit the like and subscribe button for more information and free live coaching for GATE 2023 exam preparation. BYJU’S Exam Prep GATE (Formerly Gradeup GATE) is a one-stop solution for all Engineering Exams, including online GATE coaching for various Engineering branches. Subscribe to this channel to access video lectures, GATE LIVE classes, exam analysis, post gate guidance, exam notifications, and get updates on Exam Patterns, Syllabus, Results, Tips, and free GATE online coaching and Strategy to crack GATE 2023 with flying colors. Also, prepare for PSU recruitment like NIELIT, UPSC ESE, BARC, ISRO, and PGEE exams. Stay tuned to our channel BYJU’S EXAM Prep GATE (BYJU’S GATE) to get the latest updates on GATE 2023 Exam, GATE Syllabus, GATE exam 2023 new updates, GATE exam preparation, online GATE all subjects classes, etc. 📚 👉🏼 Get Free YouTube Courses for GATE 2023, & PSU (2022 & 2023) Exam Preparation 👉GATE 2023 | CSE | Free YouTube Course | BYJU’S GATE👉https://bit.ly/3J9vMVu ***************************************************************** 📚Click Here for Free Online Coaching for GATE 2023, & PSU (2022 & 2023) Exam Preparation 📚 👉🏼 CSE | BYJU’S Exam Prep | GATE/ESE/PSU: 👉🏼 https://bit.ly/3hmguR9 ************************************************************************************************************ 📚✅GATE 2023, ESE 2022 & PSU (2022 & 2023) Exams Preparation Strategy 👉https://bit.ly/3KBJJw3 ************************************************************************ ✅📲 Subscribe to our YouTube Channel and press the 🔔 bell icon to get notified of all new videos👉 https://bit.ly/3rBxAiK ✅📱 For Hindi Content follow us on YouTube👉 https://bit.ly/3oLyYhV ************************************************************************* ​✅📲 To get PDFs of YouTube sessions, Join our Telegram Group: ✅GATE & ESE Telegram👉 https://bit.ly/3JYjIXj_TG ✅GATE CS - https://bit.ly/35szpag_CS ********************************************************************** ✅📱Follow our GATE Instagram PAGE👉 https://bit.ly/3HxpSMh_IG ✅📱Follow our GATE Facebook PAGE👉 https://bit.ly/3K5yYBV_FB ✅📲Visit Our Website👉 https://bit.ly/3stjUrS ✅📲 Download BYJU’S Exam Prep app for GATE Exam👉 https://bit.ly/3hhxxnx ******************************************************************** Subscribe to this channel to access video lectures, LIVE sessions, exam analysis, post gate guidance, exam notifications and get updates on Exam Patterns, Syllabus, Results, Tips, and Strategy. Also prepare for PSU recruitment exams, NIELIT, UPSC ESE, BARC, ISRO, and PGEE exams. #byjusgate #byjusexamprepgate #byjusgateprep #gateprep #gate2023 #gateexam #gatepreparation #gatecoaching #gateonlinecoaching #gatecomputerscience #gatecse #datastructure

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Un Árbol AVL es un tipo de Árbol Binario de Búsqueda (ABB) auto-balanceado. Esto significa que, después de cada operación (como insertar o eliminar un nodo), el árbol realiza rotaciones para mantener su altura lo más equilibrada posible. Este balanceo asegura que las operaciones de búsqueda, inserción y eliminación se mantengan siempre muy eficientes, evitando que el árbol se "degenere" y se vuelva lento.

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AVL Tree is an essential topic under one of the most important chapters of Computer Science i.e. Data Structure. And, when it comes to a competitive examination like GATE, you have to dive deep into this topic to understand it thoroughly. In this article, we have comprised all the pointers related to the AVL Tree. We hope these notes for CSE topics will help you understand this topic in a better way.
Table of Contents
The term AVL tree was introduced by Adelson-Velsky and Landis. It is a balanced binary search tree and is the first data structure like this. In the AVL tree, the heights of the subtree cannot be more than one for all nodes.

The above picture is of an AVL tree. Here, it is clearly visible that the heights of the left and right subtrees are equal to, or less than one.

The above tree is not an AVL tree. And, the reason is simple. Here, the heights of the left and right subtrees are higher than 1.

Balance Factor in AVL Tree

In the AVL tree, the term balance factor is very important.

Balance Factor = height(left-subtree) - height(right - subtree)

The balanced factor should be -1, 0 or +1. Otherwise, the tree will be considered an unbalanced tree.

Example Of AVL Tree & Balance Factor:

In the above example, the height of the left subtree is 3 and the height of the right subtree is 2. That means the balance factor is <=1 therefore the tree is supposed to be balanced.

Why Do We Need an AVL Tree?

To reduce the issue of time complexity in a binary search tree, the AVL tree was introduced by Adelson-Velski & Landis. It is a self-balancing tree that helps in reducing the complexity issue.

Operations on AVL Tree

The AVL tree is a balancing binary tree, and therefore it follows the same operations we perform in the binary search tree.

Insertion: The process of insertion is the same as it is executed in the binary search tree. However, there are chances that it may point to a violation in the AVL tree property, and the tree may require balancing. To balance a tree we can apply rotations.

Deletion: The process of deletion is the same as it is executed in a binary search tree. It can affect the balance factor of the tree, therefore, we need to utilize different types of rotations to balance the tree.

AVL Rotation

Left rotation

Right rotation

Left-Right rotation

Right-Left rotation

Left Rotation: When we perform insertion at the left subtree, then it is a left rotation.

Right Rotation: When we perform insertion at the right subtree, then it is a right rotation.

Right-Left Rotation

Figure: Right-Left Rotation

Time Complexity in AVL Tree

Practice Problem - AVL Tree

Q. What is the maximum height of any AVL-tree with 7 nodes? Assume that the height of a tree with a single node is 0.

Keep learning and stay tuned to get the latest updates on GATE Exam along with GATE Eligibility Criteria, GATE 2023, GATE Admit Card, GATE Syllabus for CSE (Computer Science Engineering), GATE CSE Notes, GATE CSE Question Paper, and more.
Also Explore,

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AVL Tree is an essential topic under one of the most important chapters of Computer Science i.e. Data Structure. And, when it comes to a competitive examination like GATE, you have to dive deep into this topic to understand it thoroughly. In this article, we have comprised all the pointers related to the AVL Tree.

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Un Árbol B es un árbol auto-balanceado optimizado para grandes volúmenes de datos en disco. Sus nodos manejan múltiples claves e hijos, minimizando los accesos a disco, lo que lo hace ideal para la indexación en bases de datos y sistemas de archivos.

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A B-tree is a sort of self-balancing search tree whereby each node could have more than two children and hold multiple keys. It's a broader version of the binary search tree. It is also usually called a height-balanced m-way tree.

In this article, we will dive deeper into B Tree according to the GATE Syllabus for (Computer Science Engineering) CSE. Keep reading ahead to learn more.
Table of Contents
What is B Tree in Data Structure?

The B Tree is a particular m-way tree which can be used to access discs in a variety of ways. At most m children and m-1 keys can be found in a B-Tree of order m. One of the main advantages of the B tree is its capacity to store a large number of keys inside a single node and huge key values while keeping the tree's height low.

Need of a B Tree in a Data Structure

The need for B-tree emerged as the demand for faster access to physical storage media such as a hard disk grew. With a bigger capacity, backup storage devices were slower. There was a demand for data structures that reduced the number of disc accesses.

A binary search tree, an AVL tree, a red-black tree, and other data structures can only store one key in each node. When you need to store a significant number of keys, the height of such trees grows quite vast, and the time it takes to access them grows.

B-tree, on the other hand, can store multiple keys inside a single node and has several child nodes. It considerably reduces the height, allowing for speedier disk access.

B Tree Example

Properties of B Tree in DBMS

All of the features of a M way tree are present in a B tree of order m. Additionally, it has the following features:

In a B-Tree, each node has a maximum of m children.

Except for the root and leaf nodes, each node in a B-Tree has at least m/2 children.

There must be at least two root nodes.

The level of all the leaf nodes should be the same.

It is not required that all nodes have the same number of children, but each node must include m/2 nodes.

The following graphic depicts a B tree of order 4:

Any attribute of B Tree may be violated while performing some operations on it, such as the number of minimum children each node can have. The tree may split or unite in order to keep the features of the B Tree.

Operations

Search Operation

B Trees are comparable to Binary Search Trees in terms of searching. For example, let's look for item 49 in the B Tree below. The procedure will go somewhat like this:

Compare item 49 to node 78, which is the root node. Move to the left sub-tree since 49 < 78.

Since 40 < 49 < 56, move to the right subtree of 40. If 49>45, move to the right, and then compare and contrast 49.

Return if a match is found.

The height of the tree affects the search in a B tree. To search an element in a B tree, the search technique requires O(log n) time.

Insertion Operation

At the leaf node level, insertions are made. In order to place an item into B Tree, the following algorithm must be followed:

Navigate the B Tree until you find the suitable leaf node to insert the node at.

If there are fewer than m-1 keys in the leaf node, insert the elements in ascending order.

Otherwise, if the leaf node comprises m-1 keys, proceed to the next step.

In the rising sequence of elements, add the new element.

In the middle, divide the node into two nodes.

The median element should be pushed up to its parent node.

If the parent node has the same m-1 number of keys as the child node, split it as well using the very same steps.

Example

In the B Tree of order 5, insert node 8 as indicated in the diagram.

Because 8 will be put to the right of 5, enter 8.

There are now 5 keys in the node, which is more than (5 -1 = 4) keys. As a result, divide the node from the median, which is number 8, and push it up to its parent node, as illustrated below.

Deletion Operation

At the leaf nodes, deletion is also conducted. It can be a leaf node or an inside node that needs to be destroyed. In order to delete a node from a B tree, use the following algorithm:

1. Determine the location of the leaf node.

2. If the leaf node has more than m/2 keys, delete the required key from the node.

3. If the leaf node lacks m/2 keys, fill in the gaps with the element from the eighth or left sibling.

If the left sibling has more than m/2 elements, shift the intervening element downwards to the node wherever the key is deleted and push the largest element back to its parent.

If the right sibling has more than m/2 items, shift the intervening element downwards to the node wherever the key is deleted, then push the smallest element up to the parent.

4. Create a new leaf node by merging two leaf nodes and the parent node's intervening element if neither sibling has more than m/2 elements.

5. If the parent has less than m/2 nodes, repeat the operation on the parent as well.

If the node to be destroyed is an internal node, its in-order successor or predecessor should be used instead. The process will be the same as the node is deleted from the leaf node because the successor/predecessor would always be on the leaf node.

Example 1

Remove node 53 from B Tree of the order 5, as indicated in the diagram below:

Element 49's right child contains the number 53. Remove it.

Now that 57 is the only element left in the node, the minimal number of elements required in a B tree of rank 5 is 2. Because the elements in its left and right subtrees are similarly insufficient, combine it with the left sibling of the intervening element of the parent, i.e. 49.

The following is the final B tree:

B Tree Application

Because accessing values held in a huge database, saved on a disc, is a very time-consuming activity, the B tree is used to index the data and enable quick access to the actual information stored on the discs.

In the worst situation, searching an unindexed and unsorted database with n key values takes O(n) time. Also, if we use B Tree for indexing this DB, it will be searched in about O(log n) time.

Practice Problems on B Tree

1. The B-trees of the order 4 and height 3 would have a maximum of ________ keys.

a. 188

b. 127

c. 63

d. 255

Answer- (d) 255

2. If five node splitting operations occur when some entry is inserted in a B-tree, how many nodes would be written?

a. 5

b. 11

c. 7

d. 14

Answer- (b) 11

3. AVL tree and B-tree consist of a similar worst case time complexity for deletion and insertion.

a. True

b. False

Answer- (a) True

4. 2-3-4 trees refer to the B-trees of order 4. These are an isometric of the _______ trees.

a. AA

b. AVL

c. Red-Black

d. 2-3

Answer- (c) Red-Black
Also Explore,

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A B-tree is a sort of self-balancing search tree whereby each node could have more than two children and hold multiple keys. It's a broader version of the binary search tree. It is also usually called a height-balanced m-way tree. In this article, we will dive deeper into B Tree according to the GATE Syllabus for (Computer Science Engineering) CSE.

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Un Árbol B+ es un índice de árbol optimizado para bases de datos (DBMS). Solo las claves de búsqueda van en nodos internos; todos los datos (o sus punteros) están en los nodos hoja, que además están enlazados. Esto maximiza la eficiencia en búsquedas de rango y minimiza accesos a disco.

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A B+ Tree is a more advanced self-balancing tree. In this, all the values are present at the leaf level. B+ Tree in the data structure is a B Tree enhancement that enables faster insertion, deletion, and search operations.

In this article, we will dive deeper into B+ Tree according to the GATE Syllabus for (Computer Science Engineering) CSE. Keep reading ahead to learn more.
Table of Contents
What is B+ Tree in DBMS?

A balanced binary search tree is the B+ Tree. It uses a multilevel indexing system. Leaf nodes in the B plus tree represent actual data references. The B plus tree keeps all of the leaf nodes at the same height. A link list is used to connect the leaf nodes in the B+ Tree. As a result, a B+ Tree can allow both random and sequential access.

Multilevel indexing is a crucial topic to grasp before understanding B+ Tree in the data structure. The index of indices is formed in multilevel indexing, as shown in the diagram below. It facilitates and accelerates data access.

B+ Tree Properties

The leaves are all at the same height.

There are at least two children in the root.

Except for root, each node can have a max of m children and a min of m/2 children.

A max of m - 1 keys and a min of m/2 - 1 keys can be stored in each node.

Both keys and records can be placed in the internal and leaf nodes of a B Tree. In a B+ Tree, records or data can only be kept on the leaf nodes, whereas key values can only be placed on the internal nodes.

To make search queries more efficient, the leaf nodes of the B plus tree in the data structure are connected together in the form of singly-linked lists.

B+ Trees are used to store vast amounts of data that are too large to fit in the main memory. The internal nodes of the B+ Tree (the keys to access records) are securely stored memory, whereas leaf nodes are placed in the secondary memory due to the restricted amount of main memory.

B plus tree internal nodes are often referred to as index nodes. The following diagram depicts a B+ Tree of order 3.

Pros of B+ Tree

In an equal number of disc accesses, records can be retrieved.

In comparison to the B tree, the B+ Tree's height stays balanced and is lower.

The stored data in a B+ Tree can be accessed both sequentially and directly.

Indexing is done via keys.

Because the data is only stored on the leaf nodes, search queries are faster.

B+ Tree Structure

Every leaf node in the B+ Tree is at the same distance from the root node. The B+ Tree has an order of n, with n being the same for all B plus trees.

It has a leaf node and an internal node.

Internal Node

Except for the root node, an internal node present in the B+ Tree can at least have n/2 record pointers.

The internal node of the tree can only have n pointers.

Leaf Node

At least n/2 key values and n/2 record pointers can be found in the B plus tree's leaf node.

A leaf node can only have n record pointers and n key values.

Every B+ Tree leaf node has a block pointer P that points to the next leaf node.

Searching Records in B+ Tree

Suppose that we need to find 55 in the B+ Tree structure below. We'll start by fetching for the intermediary node, which will lead to the leaf node, which may have a record of 55.

So we'll identify a branch between 50 to 75 nodes in the intermediary node. Then we'll be redirected to the 3rd leaf node at the conclusion. In this case, the DBMS will use a sequential search to locate 55.

Insertion in B+ Tree

Suppose that we wish to add a record 60 to the structure below. After 55, it will reach the third leaf node. It's a balanced tree, and one of its leaf nodes is already full; thus, we can't put 60 in there.

We must split the leaf node in this scenario so that it may be added to the tree without impacting the fill factor, balance or order.

The values of the 3rd leaf node are (50, 55, 60, 65, 70), and the current root node is 50. To keep the tree's equilibrium, we'll divide the leaf node in half in the middle. As a result, (50, 55) & (60, 65, 70) can be grouped into two leaf nodes.

The intermediate node can't branch from 50 if these two must be leaf nodes. It must have 60 added to it, and thereafter links to a new leaf node will be available.

When there is an overflow, we can enter an entry in this manner. In a typical scenario, finding the node in which it fits and then placing it in that leaf node is extremely simple.

Deletion in B+ Tree

Imagine that we want to remove 60 from the previous example. In this scenario, we must subtract 60 from both the intermediate node and the fourth leaf node. The tree will not satisfy the B plus tree rule if it is removed from the intermediate node. As a result, we must alter it in order to achieve a balanced tree.

It will look like the following after eliminating node 60 from the B+ Tree and rearranging the nodes:

Practice Problems on B+ Tree in Data Structure

1. What's the max number of keys in a B+ Tree of height 3 and order 3?

a. 27

b. 3

c. 26

d. 80

Answer- (c) 26

2. Which of these data structures would be preferred in DB-system implementation?

a. B Tree

b. AVL Tree

c. Splay Tree

d. B+ Tree

Answer- (d) B+ Tree

3. In the case of a B+ Tree, both the leaves and the internal nodes consist of keys.

a. False

b. True

Answer- (a) False

4. What would be the min number of keys in the leaves in case any B+ Tree would contain a max of 7 pointers in the node?

a. 7

b. 3

c. 6

d. 4

Answer- (b) 3

Keep learning and stay tuned to get the latest updates on the GATE Exam along with Eligibility Criteria, GATE Syllabus for CSE (Computer Science Engineering), GATE CSE Notes, GATE CSE Question Paper, and more.
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A B+ Tree is a more advanced self-balancing tree. In this, all the values are present at the leaf level. B+ Tree in the data structure is a B Tree enhancement that enables faster insertion, deletion, and search operations. In this article, we will dive deeper into B+ Tree according to the GATE Syllabus for (Computer Science Engineering) CSE.

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