Question

Prove that any tree with two or more vertices has at least two vertices of degree one.

Short answer

A tree with two or more vertices always has at least two leaves, i.e., vertices of degree one.

Step-by-step solution

Understanding a Tree

A tree is a connected graph with no cycles. In other words, it is a collection of vertices (nodes) and edges (connections) where any two nodes are connected by exactly one path. Trees are known for having $n-1$ edges, where $n$ is the number of vertices. Identifying properties like these is crucial for proofs.

Identify a Path in the Tree

Assume you have a tree with two or more vertices. A path is a sequence of vertices such that each consecutive pair is connected by an edge in the graph. Let's consider a path in the tree that has the maximum length, known as a diameter path.

Examine the End Vertices

Notice the end vertices of this longest path – these are called the leaves of the tree. In any graph, and particularly in trees, a vertex with only one connecting edge is a leaf, having a degree of one.

Consider the Structure of a Tree

Because the path is the longest, there cannot be any additional edges connected to these end vertices. If there were, this would create a longer path, contradicting the path's status as the longest. Therefore, these end vertices must have only one edge connecting them to the rest of the tree.

Conclude with Two Leaves

Since the tree had at least two vertices to start with and we considered the longest path, the end vertices of this path must be those two vertices of degree one. Hence, every tree with two or more vertices must have at least two vertices of degree one.

Key concepts

Tree Graphs

Tree graphs are fascinating structures in graph theory that model hierarchical organizations excellently. A tree is a type of graph that is uniquely characterized by its lack of cycles, meaning that you can start at one vertex and follow edges to any other vertex without looping back to the original. Imagine having nodes connected in layers like a family tree or an organizational chart.
Trees are connected graphs, so there is exactly one path connecting any two vertices. This lack of multiple paths is what distinguishes trees from other types of connected graphs. A tree with $n$ vertices always has $n-1$ edges. This is a cornerstone property for understanding and proving other aspects of tree graphs.

• Connectedness: Every vertex in a tree can be reached from any other vertex.
• Cycle-Free: Trees avoid redundancy by ensuring there are no closed loops.
• Edge Count: The number of edges in a tree graph is always one less than the number of its vertices.

Vertex Degree

The vertex degree in a graph is an essential concept signifying the number of edges that connect to a vertex. In tree graphs, which have their own unique properties, the concept of vertex degree helps determine and classify vertices, specifically leaves and internal nodes.
Vertices in a graph can be:

• **Leaf:** A vertex with degree one, meaning it is a terminal vertex.
• **Internal Vertex:** Any vertex with a degree of two or more, typically situated between two or more connected paths.

Each tree will always have at least two leaves. These terminal points signify where paths begin or end and are crucial for verifying the properties of the tree, like in the proof provided in the exercise.

Graph Path

A graph path is a sequence of vertices where each adjacent pair is connected by an edge. In trees, paths represent the simplest route from one vertex to another without returning or looping back on previous vertices.
The longest path in a tree is called the diameter. It spans through the most significant number of varying vertices and edges, acting as a backbone of the tree's structure. This is not only important for characterizing the structure but also explains why the end vertices of this path, intuitively, are leaves, having only one connecting edge.

• **Path Length:** The number of edges in a path.
• **Diameter of a Tree:** The longest path length in the tree.
• **Endpoints of Path:** The vertices present at the ends of the longest path, or the leaves.

Cycle-Free Graphs

Cycle-free graphs are exactly what they sound like: graphs that do not contain any cycles. In graph theory, a cycle is a path that begins and ends at the same vertex without retracing any edge.
Trees are prime examples of cycle-free graphs. They are structured such that you can only move from one vertex to another through a single, one-way path. This lack of cycles ensures data or resource flow without confusion or duplication, which is why trees are integral in computing algorithms and management schemas.

• **Cycle Definition:** A sequence where the starting and ending vertex is the same, forming a loop.
• **Cycle-Free Implication:** Guarantees no loops, ensuring efficient traversal and data paths.
• **Impacts on Trees:** Confirming each tree must have a minimum number of leaves.

Understanding cycle-free properties in tree graphs helps to fully appreciate why such graphs are highly organized and predictable, aiding both theoretical proofs and practical applications.