Trees and Spanning Trees is a part of the VCE Further Maths topic Networks and Decision Mathematics. It is part of the subtopic Trees and Minimum Connector Problems.
A tree is a connected graph that contains no cycles, multiple edges or loops. A spanning tree represents a connection between all vertices without certain edges which are usually less-optimal in length.
What is a Tree?
This next video is a great introduction to trees and spanning trees.
A tree is a connected undirected graph that contains no cycles. It is a mathematical representation of a hierarchical structure where each vertex, also known as a node, is connected to one or more other vertices.
A tree can be visualised as a branching structure, where each node has one parent and zero or more children. The topmost node is called the root and the nodes that are not connected to any other node are called the leaves.
A spanning tree of a connected graph is a subgraph that includes all the vertices of the original graph and is also a tree. In other words, it is a tree that spans all the vertices of the original graph, but with some of the edges removed.
The process of removing edges to get a spanning tree is called pruning, and it is used to simplify the structure of a graph while still preserving its connectivity.
In graph theory, the study of trees and spanning trees is important because they can be used to model and analyse various real-world systems, such as networks of roads, electrical circuits, and social connections. They can also be used to find the shortest paths between nodes, to group nodes into clusters, and to study the properties of the graph such as connectivity, Euler’s theorem, and Hamiltonian cycles.
Understanding the definitions and properties of these concepts is crucial for the study of graph theory and its applications in various fields such as computer science, operations research, and network analysis.
Want to learn more? Check out more of our VCE Mathematics resources here!
