Spanning Tree vs Minimum Spanning Tree

Spanning Tree (ST):

A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and is also a tree (a connected acyclic graph).

The primary goal of a spanning tree is to connect all vertices with the minimum number of edges.

Uses of Spanning Tree:

• STs are used in network design and routing algorithms to ensure connectivity without cycles.
• They help identify bridges and critical connections in a network.
• Spanning trees are also used in data structures like Disjoint-Set Union (Union-Find) for efficient cycle detection.

Minimum Spanning Tree (MST):

A minimum spanning tree of a weighted graph is a spanning tree with the minimum possible sum of edge weights.

MST is used when the graph is weighted, and the goal is to find the subset of edges that connects all vertices with the minimum total weight.

Use of MST:

• MSTs have applications in network design, such as in constructing efficient communication networks with the minimum total cost.
• They are used in clustering and hierarchical data structures.
• MST algorithms are applied in image segmentation, clustering, and minimum spanning tree-based approximations for optimization problems.

Differences between Spanning Tree (ST) and Minimum Spanning Tree (MST):

Category	Spanning Tree (ST)	Minimum Spanning Tree (MST)
Objective	Connect all vertices with minimum edges.	Connect all vertices with minimum total edge weight.
Graph Type	Can be used with both weighted and unweighted graphs.	Primarily used with weighted graphs.
Edge Weight	Edge weight is not a consideration.	Edge weight is the key consideration.
Algorithms	No specific algorithm for ST. Often used in algorithms like DFS and BFS.	Algorithms like Kruskal’s or Prim’s are used to find MST.
Applications	Used for network connectivity and cycle detection.	Applied in network design, clustering, and optimization problems with cost constraints.
Construction Method	No unique method to construct, depends on the specific problem and context.	Algorithms explicitly designed to find the minimum spanning tree.
Edge Count	The number of edges in the spanning tree is not optimized for minimum.	The number of edges is minimized to connect all vertices.