Problem p02752 - Generation 2

Orig Description

Score : 1100 points
Problem Statement
We have a tree G with N vertices numbered 1 to N.
The i-th edge of G connects Vertex a_i and Vertex b_i.
Consider adding zero or more edges in G, and let H be the graph resulted.
Find the number of graphs H that satisfy the following conditions, modulo 998244353.
H does not contain self-loops or multiple edges.
The diameters of G and H are equal.
For every pair of vertices in H that is not directly connected by an edge, the addition of an edge directly connecting them would reduce the diameter of the graph.
Constraints
 3 \le N \le 2 \times 10^5 
 1 \le a_i, b_i \le N 
The given graph is a tree.
Input
Input is given from Standard Input in the following format:
N
a_1 b_1
\vdots
a_{N-1} b_{N-1}
Output
Print the answer.
Sample Input 16
1 6
2 1
5 2
3 4
2 3
Sample Output 13
For example, adding the edges (1, 5), (3, 5) in G satisfies the conditions.
Sample Input 23
1 2
2 3
Sample Output 21
The only graph H that satisfies the conditions is G.
Sample Input 39
1 2
2 3
4 2
1 7
6 1
2 5
5 9
6 8
Sample Output 327
Sample Input 419
2 4
15 8
1 16
1 3
12 19
1 18
7 11
11 15
12 9
1 6
7 14
18 2
13 12
13 5
16 13
7 1
11 10
7 17
Sample Output 478732

Extracted Specification

An integer N (3 ≤ N ≤ 2 × 10^5).
N-1 pairs of integers (a_i, b_i) where 1 ≤ a_i, b_i ≤ N.

### Example Input:

```
6
1 6
2 1
5 2
3 4
2 3
```

### Function Signature:
Write a function f(N, edges) that takes in the input.
```python
def f(N: int, edges: List[Tuple[int, int]]):
    ''' 
    N: an integer 
    edges: a list of N-1 tuples, each containing two integers
    '''
```

Test Case Generator

Sure, I can help you write a test case generator for this problem. Here's the Python code that generates random test cases for the given problem statement:

```python
import random

def generate_tree_edges(N):
    edges = []
    available_nodes = list(range(2, N+1))
    random.shuffle(available_nodes)
    
    for i in range(1, N):
        parent = random.randint(1, i)
        child = available_nodes.pop()
        edges.append((parent, child))
    
    return edges

def tcgen():
    N = random.randint(3, 200000)
    
    edges = generate_tree_edges(N)
    
    return N, edges

def print_test_case():
    N, edges = tcgen()
    print(N)
    for edge in edges:
        print(edge[0], edge[1])

# Example usage
print_test_case()
```

### Explanation:
1. **generate_tree_edges(N)**: This function generates edges for a tree with \(N\) nodes. It ensures that the tree remains connected by connecting each new node to a random existing node.
2. **tcgen()**: This function generates a random \(N\) (number of nodes) and calls `generate_tree_edges` to get the edges of the tree.
3. **print_test_case()**: This function prints the generated test case in the required format.

### Steps:
1. **Randomly select \(N\)**: The number of vertices in the tree.
2. **Generate Tree Edges**: Ensure a connected tree by always connecting each new node to an existing node.
3. **Print the Test Case**: Format the output to match the required input format.

You can run the `print_test_case()` function to generate and print a random test case for the problem. Adjust the random seed or modify the functions as needed to get specific types of test cases.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    edges = []
    for _ in range(N - 1):
        a, b = map(int, fh.readline().strip().split())
        edges.append((a, b))
    return N, edges