Question

Draw (1) all trees with 4 vertices. (2) all trees with 8 vertices.

Short answer

The short answer for all trees with 4 vertices are: 1. A star tree with a central vertex connected to three leaves. 2. A straight line of 4 vertices. 3. A central vertex connected to two leaves and a straight line of 2 vertices connected. For all trees with 8 vertices, due to the complexity and large number of tree structures, it is recommended to use a computer program or online tools like GraphTheory on Wolfram Alpha or IGraph to generate them. Then, study and reproduce the unique tree structures to complete the exercise.

Step-by-step solution

Part 1: All Trees with 4 vertices

For this part of the exercise, we will find all trees with 4 vertices: Step 1: Start with a tree of one vertex Draw a single vertex. This is the starting point for creating trees with more vertices. Step 2: Add the second vertex Add another vertex and connect it to the first vertex to form a tree of 2 vertices connected by an edge. Step 3: Add the third vertex Add a new vertex and consider all the possible new connections. There are two possibilities: - Connect the new vertex to the first vertex; tree structure: a central vertex with two leaves. - Connect the new vertex to the second vertex; tree structure: a straight line of 3 vertices. Step 4: Add the fourth vertex For each of the two trees found in Step 3, add a new vertex and find all the possible connections. Remember to avoid any cycles and ensure that the tree remains connected. For the first tree structure from Step 3 (central vertex with two leaves): - Add the fourth vertex and connect it to the central vertex; tree structure: star tree with a central vertex connected to three leaves. For the second tree structure from Step 3 (straight line of 3 vertices): - Add the fourth vertex and connect it to one of the end vertices; tree structure: a straight line of 4 vertices. - Add the fourth vertex and connect it to the middle vertex; tree structure: a central vertex with two leaves and a straight line of 2 vertices connected. All three tree structures are unique. The final trees with 4 vertices are: 1. A star tree with a central vertex connected to three leaves. 2. A straight line of 4 vertices. 3. A central vertex connected to two leaves and a straight line of 2 vertices connected.

Part 2: All Trees with 8 vertices

For this part of the exercise, we will find all trees with 8 vertices. Due to the large number of tree structures, it's more efficient to use a systematic approach based on the existing smaller trees found in the first part. Step 1: Start with the trees from Part 1 We already have three different tree structures with 4 vertices. Each structure will serve as a starting point for creating larger trees with 8 vertices. Step 2: Add vertices to each tree structure For each of the three tree structures from Part 1, add vertices and find all the possible connections when each vertex is added in such a way that no cycles are formed while keeping the tree connected. Repeat this process until each tree has 8 vertices. Step 3: Identify unique tree structures After creating all the possible trees with 8 vertices from the tree structures with 4 vertices, compare them to identify and discard any repeated trees, leaving only the unique tree structures. Due to the complexity of this part of the exercise and the number of possible tree structures, we recommend using a computer program or an online tool to generate the trees. Examples of online tools that can be used to generate trees with 8 vertices include the following: 1. GraphTheory on Wolfram Alpha 2. IGraph: an open-source network analysis software Once the trees are generated, the student can study and reproduce the unique tree structures to complete the exercise.

Key concepts

Vertex

A vertex is a fundamental unit in graph theory. Think of it as a dot that can connect with other dots. In the context of trees, a very specific type of graph, a vertex serves as a point where you can connect lines, known as edges.

• Vertices are the basic entities that form the network in a graph.
• In tree structures, vertices represent nodes that are interconnected without any closed loops.

In the exercise you are working on, trees with 4 or 8 vertices need to be created. This means you will lay down dots (vertices) and connect them with lines (edges) to form a network where all the points are linked together without forming cycles. Each added vertex brings new possibilities for connections, which allows you to explore different structures of trees.
Understanding how vertices relate and connect with one another is essential in creating tree-like structures, which helps solve the problem in the exercise efficiently.

Graph Structure

A graph structure is a mathematical representation that consists of a set of vertices and edges. Trees are a special kind of graph with some unique properties. They are acyclic, meaning they do not contain cycles. No vertex is revisited, once you travel through the connections.

• Each graph structure in our exercise is a tree that must hold these properties.
• A tree with \( n \) vertices will have exactly \( n-1 \) edges.
• Graph structures can vary in shape and form, such as linear paths or more branched out formations.

In the task of drawing all trees with a given number of vertices, understanding and utilizing these properties of graph structures is crucial. For instance, in trees with 4 vertices, these can include simple layouts like a star shape or a straight line, depending on how the vertices are connected. It's key to ensure all the edges link vertices in a certain way to form a feasible tree structure adhering to the foundational principles of graph theory.

Discrete Mathematics

Discrete mathematics is the foundation of graph theory, providing the theoretical groundwork for understanding trees and other graphs. It deals with discrete elements unlike continuous mathematics. It includes the study of graphs, networks, and other structures involving discrete components.

• It includes topics like logic, set theory, and, importantly for this exercise, graph theory.
• Understanding discrete math helps in visualizing how different vertices and edges interact in a graph structure.

The exercise leverages principles from discrete mathematics to help find all possible tree structures with a specific number of vertices. Graph theory, a subfield of discrete mathematics, gives us the tools and methods to systematically explore the relationships and connections between different vertices and edges, forming trees. Appreciating the discrete nature of trees helps in solving graph-related problems and building a comprehensive understanding of how graphs function in logical and algorithmic contexts.