Question

Construct all trees on six vertices. Find an algorithm for constructing all possible trees on six vertices if you know all possible trees on five vertices.

Short answer

There are 30 possible trees on six vertices.

Step-by-step solution

Understanding the Problem

We are asked to find all trees with six vertices. We will begin by considering trees with one less vertex, i.e., trees with five vertices, and then use these to find trees with six vertices.

Definition Review

Recall that a tree is a connected graph with no cycles. For a tree with n vertices, it has exactly n-1 edges. Thus, a tree with six vertices has five edges.

Start from Five Vertices

List all possible tree structures with five vertices. There are six different tree structures possible with five vertices, which we will use as a base to construct trees with six vertices.

Algorithm for Adding a Vertex

To construct a tree with six vertices from trees with five vertices, add one new vertex to each possible tree with five vertices and create an edge between this new vertex and each of the existing vertices.

Implementation for Six-Vertex Trees

Apply the algorithm: Add a sixth vertex to each position on the existing five-vertex trees. Essentially, for each of the six trees with five vertices, connect the new vertex to each of the five vertices, resulting in five new trees per original tree.

Count of Six-Vertex Trees

Since each of the six trees with five vertices can generate five new trees by adding a sixth vertex, there are sufficient combinations to ensure all unique large trees. Calculate: 6 (five-vertex trees) x 5 (connections per tree) = 30 possible six-vertex trees.

Key concepts

Graph Theory

Graph theory is a field of mathematics that studies graphs. A graph is a collection of vertices (or nodes) and edges that connect pairs of vertices. There are different types of graphs, such as trees, which are particularly important structures in graph theory.

A tree is a special kind of graph that is connected, meaning there is a path between any two vertices, and acyclic, meaning it doesn’t have closed loops. This property makes trees easy to work with in algorithmic applications, as they represent hierarchical structures, like family trees or organizational charts.

Understanding graph theory helps in designing algorithms to solve problems such as finding the shortest path in a network or creating minimal spanning trees. In our problem, we focus on constructing trees with six vertices based on knowledge of trees with five vertices, a task rooted in graph theory's foundational concepts.

Vertex Addition

When constructing trees, vertex addition is a process where a new vertex is introduced into the existing graph structure. In simpler terms, you start with a graph (a tree, in this case) and add more vertices one at a time.

In our exercise, we begin with trees consisting of five vertices. For every possible configuration of these trees, a new vertex is added. This vertex is then connected to each existing vertex, ensuring that the new structure remains a tree. By connecting the new vertex to existing ones, you create different six-vertex trees.

Key considerations during vertex addition include:

• A tree with $n$ vertices must have $n-1$ edges to remain connected and acyclic.
• Each addition must not create cycles, maintaining the definition of a tree.

By systematically employing this method, you ensure every possible tree structure with six vertices is accounted for.

Tree Algorithm

Tree algorithms are methods used to create and manipulate tree data structures. In this context, our task is to utilize an algorithm that allows expanding trees from five to six vertices.

The algorithm works as follows:

• Start with all configurations of five-vertex trees.
• Add one additional vertex to each configuration.
• Create an edge from this new vertex to each vertex in the five-vertex trees.

This approach results in five different positions for placing the new vertex for each existing tree configuration, creating five new possibilities. This step is crucial for exploring all potential six-vertex trees without omission. Ultimately, it acts as a blueprint that guides the growth of smaller tree structures into larger ones using systematic vertex addition.

Combinatorial Structures

In mathematics, combinatorial structures involve arrangements and relationships within a set or group, like permutations, combinations, and graph configurations.

A tree with six vertices is one such combinatorial structure. It involves specific arrangements of vertices to ensure that the defining properties of trees are met. In the exercise, constructing all possible trees with six vertices involves understanding how these trees can form distinct patterns based on smaller configurations (five-vertex trees).

By understanding combinatorial principles, we know that the number of distinct trees we create is governed by the various ways we can arrange and connect vertices. In this exercise, starting with six configurations of trees with five vertices and adding connections systematically to each with a new vertex, we devise 30 possible arrangements. This highlights the application of combinatorial logic to ensure comprehensive coverage of all potential structures. The results not only show how diverse structures can stem from simple ones but also illustrate the versatility and adaptability of trees as a combinatorial concept.