To construct a graph that is a tree with vertices $P$, $Q$, $R$, $S$, and $T$, where the degree of $S$ is 3, let's follow these steps:

Step 1: Understanding the properties of a tree

A tree is a connected, acyclic graph. For a tree with $n$ vertices, it must have exactly $n-1$ edges. In this case, we have 5 vertices (P, Q, R, S, and T), so the tree must have 4 edges.

Step 2: The degree of $S$ is 3

The degree of a vertex is the number of edges connected to it. Since the degree of $S$ is 3, it must be connected to 3 other vertices. Therefore, we need to connect $S$ to three other vertices.

Step 3: Choosing the other connections

After ensuring $S$'s degree is 3, we have 2 remaining vertices: $P$, $Q$, $R$, and $T$ to choose from. Let's connect these remaining vertices to ensure the tree remains connected and acyclic.

Step 4: Construct the tree and edge set

We can arrange the tree as follows:

• Connect $S$ to $P$, $Q$, and $R$ to fulfill the degree condition for $S$.
• Then, connect the remaining vertices to make sure the graph is connected, and no cycles exist.

Let's connect $R$ to $T$, which ensures there is one more edge and no cycles are created.

Thus, the edge set of the tree is:

$E = \{ (S, P), (S, Q), (S, R), (R, T) \}$

Final Tree Diagram:

     P
     |
     S - Q
     |
     R - T

Edge Set:

$E = \{ (S, P), (S, Q), (S, R), (R, T) \}$

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Let me know if you'd like more details or have any questions! Here are a few related questions for further exploration:

1. How do you prove that a given graph is a tree?
2. What is the maximum number of edges a tree with $n$ vertices can have?
3. How does the degree of a vertex relate to the structure of the tree?
4. Can a graph with 4 edges and 5 vertices have a cycle?
5. What is the difference between a tree and a forest in graph theory?

Tip: The degree of a vertex is crucial when constructing graphs and trees. Always check the degree constraints before adding edges.