give answer in 2 step with explanation at the end of each step and final answer at the end:
Finally, let's count some trees! A little vocabulary will be necessary here. A LABELING of a graph
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Question

give answer in 2 step with explanation at the end of each step and final answer at the end: 
Finally, let's count some trees! A little vocabulary will be necessary here. A LABELING of a graph 
�
 ﻿is a function from the vertices of 
�
 ﻿to the natural numbers, L: V
-
>
 ﻿N
.
 ﻿Two labelings are DISTINCT when the labeled graphs are nonisomorphic. As an example, the left graph below is distinct from the middle graph because the center node is labeled 
3
 ﻿instead of 
2
 
.
 ﻿However, the middle graph is NOT distinct from the right graph, which is simply the middle graph flipped horizontally.
Basically, you can move the vertices around and the labeling will stay the same, but if you switch which numbers are in which vertices, then the labelings become distinct. The left and middle graphs above have distinct labelings because no movement of the vertices will change the fact that the center vertex is labeled 
3
 ﻿on the left graph and 
2
 ﻿in the middle. As a start, there is only one distinct labeled tree on one vertex, and one distinct labeled tree on two vertices. They are:
(
1
)
。
 
◻
 ﻿O
-
O
a
.
 ﻿Draw the only tree with three vertices and all three

VIPSolutions · Accepted Answer

Let's answer each part step by step with explanations after each step and **final answers** at the end.

---
### a. Draw the only tree with three vertices and all three of its distinct labelings.

#### **Step 1: Draw the tree**
There is only one nonisomorphic tree with three vertices—it is a path:  
```
O—O—O
```

#### **Step 2: List all distinct labelings**
For vertices labeled 1, 2, 3, labelings are distinct if they can't be matched via flipping (isomorphism).  
Since this is a path of three, the only symmetry is flipping the ends.  
Labelings:  
- 1—2—3  
- 1—3—2  
- 2—1—3

**Explanation:**  
Each labeling is distinct because moving (flipping) does not allow you to turn one into another unless the labels match at the ends.

---

### b. Draw the two nonisomorphic trees with four vertices and list the distinct labelings of both.

#### **Step 1: List the two nonisomorphic trees**
- **Star:** One center, three leaves (degree 3 at center)
```
    O
    |
O—O—O
```
- **Path:** Four in a line
```
O—O—O—O
```

#### **Step 2: List distinct labelings**
- **Star:**  
  Center can be labeled with any of {1,2,3,4}. The leaves are indistinguishable, so labelings only differ by which label is at the center.  
  **Number of distinct labelings:** 4

- **Path:**  
  Only symmetry is flipping the path; labelings are distinct if you can't flip to get the same labeling.  
  For 4 nodes, there are 3 distinct labelings:  
  - 1—2—3—4  
  - 1—3—2—4  
  - 2—1—3—4  
  (and so on, for a total of 3)

**Explanation:**  
Star: Only the center matters. Path: Flip for symmetry.

---

### c. Draw the three nonisomorphic trees with five vertices and say how many distinct labelings each has.

#### **Step 1: List the three trees**
- **Star:** Center with 4 leaves
- **Path:** 5 in a line
- **"Fork":** One vertex of degree 3, which is connected to one leaf and to a path of length 2

#### **Step 2: Number of distinct labelings**
- **Star:** 5 (center can be any of the 5 labels)
- **Path:** (number of labelings up to reversal) = 12
- **Fork:** More complicated, but for the "fork," the center must be labeled, and permutations of the branches are considered. **Number is 10.**

**Explanation:**  
- **Star:** Central node's label determines the labeling; leaves are indistinguishable.
- **Path:** Linear, only flip symmetry.
- **Fork:** Some symmetry due to the branches.

---

### d. How many ways can you pick (n-2) objects from a set of n objects with repetition?

#### **Step 1: Use the formula for combinations with repetition**
The formula:  
$$
\text{Number of ways} = \binom{n + (n-2) - 1}{n-2} = \binom{2n-3}{n-2}
$$

#### **Step 2: Explanation**
This is the "stars and bars" formula for choosing (n-2) elements from n possibilities with repetition allowed.

---

### e. What is the Prüfer sequence for the tree on the right? (from the image)

#### **Step 1: Label the tree's vertices**
From the picture, the right tree is a star centered at 5, with leaves 1, 2, 3, 4, 6, 7, 8.

#### **Step 2: Build the Prüfer sequence (remove smallest leaf, record neighbor)**
The procedure is:
- Remove 1 (neighbor 5): add 5
- Remove 2 (neighbor 5): add 5
- Remove 3 (neighbor 5): add 5
- Remove 4 (neighbor 5): add 5
- Remove 5 (now neighbors with 6,7,8): add the smallest neighbor for each removal
Continue until 2 vertices left.

**Result:**  
Prüfer sequence: [5, 5, 5, 5, 5, 5]

---

### f. Draw the tree from the Prüfer sequence (1, 7, 6, 6, 1).

#### **Step 1: List labels**
Labels are from 1 to 6 (since Prüfer sequence length = n-2, so n=7).

#### **Step 2: Reconstruct**
- Sequence: [1, 7, 6, 6, 1]
- Remaining labels: 1, 2, 3, 4, 5, 6, 7

Follow the algorithm:
1. Smallest not in sequence: 2 (connect to 1), remove 2 and first 1 from sequence.
2. Smallest not in sequence: 3 (connect to 7), remove 3 and first 7.
3. Smallest not in sequence: 4 (connect to 6), remove 4 and first 6.
4. Smallest not in sequence: 5 (connect to 6), remove 5 and next 6.
5. Smallest not in sequence: 6 (connect to 1), remove 6 and next 1.
6. Last two numbers: 1, 7 (connect them).

Draw the edges:
- 2—1
- 3—7
- 4—6
- 5—6
- 6—1
- 1—7

---

### g. Using Prüfer sequences, prove Cayley's Formula: the number of distinctly labeled trees on n vertices is n^(n-2).

#### **Step 1: Count the number of Prüfer sequences**
A Prüfer sequence for n vertices is a sequence of n-2 numbers, each between 1 and n.

#### **Step 2: Each sequence corresponds to a unique tree**
So, there are n choices for each of n-2 positions: n^{n-2} unique labeled trees.

**Explanation:**  
There is a one-to-one correspondence between labeled trees and Prüfer sequences.

---

## **Final Answers**

**a.** One tree with three vertices (path), three distinct labelings (by ordering 1,2,3).

**b.** Two trees for four vertices: star (4 labelings), path (3 labelings).

**c.** Three trees for five vertices: star (5), path (12), fork (10) distinct labelings.

**d.** $\binom{2n-3}{n-2}$ ways.

**e.** Prüfer sequence for the right tree: [5, 5, 5

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