# What is a Spanning Tree in Data Structure

### Data structures

Data management is called database management. This allows the computer to sort or organize the data for efficient retrieval. A data model is a system used to store, manage, and optimize computer resources. Data processing is not just about data storage. Almost every app or program has notices and updates about its version. Data structures are so simple and complex that it is difficult to program with a programming language that does not have data structures.

Data processing is the process of using intelligence and software to manage, organize, store and store information on a computer device or system. Data models provide visualization for easy data organization and management. Any basic process, program, or program has two parts: data and algorithms - the rules and regulations of data exchange and algorithms.

There are two types of data structures

- Linear Data structures
- Non-linear data structures

**Linear Data structures**

This data type adds data to the data type. It's all about the process. You can then delete the duplicate. There are four types of linear data, they are:

- Queue
- Stack
- Linked lists
- Array

**Non-linear Data structures**

Data formats can be created in a variety of ways. There are two types of interpersonal communication:

- Tree data structure
- Graph data structure

### Tree Data structure

The information about the tree is self-explanatory. Trees are ordered and, therefore, not linear. But they are actually designed differently.

Tree A node-based data model that represents and supports the structure as a structure. In an asynchronous database, data is stored in a tree data structure called a database. All data types are stored in a central location. Each line of text is called the lower branch of the data type tree.

Two types of plants. This is limited to confidential information. Because this data processing is an individual process. So, there can be more than two children. This means that a binary plant can produce 0, 1, or 2 seeds at any time. Binary search trees can quickly parse nested and linked expressions. This allows binary trees to provide values ??from list and associate arrays. This makes it easier to find hidden items. (because it is a powerful data structure)

### Graph data structure

The graph has two sets, which are considered V and E. These vertices are also called nodes, and edges are referred to as arcs connecting any two nodes in a graph. In a simple sentence, we can say that graph- is a set of vertices (V) and edges (E) which is denoted as G (V, E). These are usually accustomed to solving several real-time problems efficiently. These generally represent networks like a city's telephones, circuitry, and paths. These are also used on many social platforms like Facebook, LinkedIn, etc. Let us consider an example of each user or person on Facebook as a vertex (or node) in the graph. Each vertex has information about the user, like name, gender, etc., and the connection between each node or vertex is considered as edges or arcs. These are also used in the operating system as resource location graphs.

These are also used in Google maps to find the shortest distance between two places, these are also used in airlines to know the best effective route, used in transportation, in circuits it is used to represent the nodes or circuitry points, these are used to solve challenging puzzles like mazes, these are also used in computer networking, there are many advantages of using graphs like it is straightforward to work with the algorithms like DFS and BFS, it has many practical applications, because of its nature(non-linear data structure) it is used in solving many complex problems, it helps in understanding and visualizing complex issues. As we know, every coin has the other side. Similarly, graphs also have a few disadvantages; they generally use many pointers, which are very difficult to manage and require large memory.

### Spanning tree

The subsets of graphs where all the vertices of the graphs are covered with the minimum number of edges that are possible are called spanning trees. Therefore, we can conclude that a spanning tree can have no cycles nor can be disconnected between any two vertices of the graph (It must be a connected graph)

Note

Connected graph

A graph that has at least one possible path with connected edges between any two selected vertices of the same graph is known as a connected graph.

Disconnected graph

A graph that has at least two adjacent vertices of the same graph that are disconnected (do not have an edge between them) is known as a disconnected graph.

Based on the definition of a spanning tree, we can observe that all the uni-directed and connected graphs have at least one spanning tree. A disconnected graph does not have any spanning tree, as a path cannot be formed between the two disconnected vertices.

Note

Uni-Directed graph

The graphs whose edges are directed in a single direction are treated as uni-direction graphs.

Cyclic graphs

The graphs whose edges form at least one cycle or loop are called cyclic graphs.

Acyclic graphs

The graphs whose edges do not form any cycles or loops are called acyclic graphs.

Therefore, it can be concluded that a single connected graph can possess more than one spanning tree. Spanning trees are subsets of the original graph, and disconnected graphs can never possess a spanning tree.

### Properties of spanning trees

Let us look at the properties of spanning trees:

- A single connected graph can possess more than one spanning tree.
- The number of edges of all the spanning trees formed from a single graph G is equal.
- Spanning trees are acyclic in nature, i.e., they do not contain any loops.
- We depict a spanning tree to be minimally connected, i.e., if an edge is removed between any two vertices of a spanning tree, then it becomes disconnected.
- We also depict a spanning tree to be minimally acyclic, i.e., if an edge is added between any two vertices of a spanning tree, then it becomes cyclic in nature.

### Mathematical Properties of spanning trees

Let us look at the mathematical properties of spanning trees:

- If a spanning tree has N number of vertices, then its number of edges is given by (n – 1)
- A spanning tree can be constructed by removing a maximum number of e – n + 1 edges from a given graph G.
- A maximum of n
^{ n – 2}number of spanning trees can be formed for a given complete graph.

### Applications of spanning trees

Spanning trees have many applications in data structures. Some of the applications are given below:

- Spanning trees are used in civil network planning
- Spanning trees are used in computer network routing protocols.
- Spanning trees are used in cluster analysis.