Orig Description
Score : 1400 points
Problem StatementWe will recursively define uninity of a tree, as follows: (Uni is a Japanese word for sea urchins.)
A tree consisting of one vertex is a tree of uninity 0.
Suppose there are zero or more trees of uninity k, and a vertex v. If a vertex is selected from each tree of uninity k and connected to v with an edge, the resulting tree is a tree of uninity k+1.
It can be shown that a tree of uninity k is also a tree of uninity k+1,k+2,..., and so forth.
You are given a tree consisting of N vertices.
The vertices of the tree are numbered 1 through N, and the i-th of the N-1 edges connects vertices a_i and b_i.
Find the minimum k such that the given tree is a tree of uninity k.
Constraints
2 ≦ N ≦ 10^5
1 ≦ a_i, b_i ≦ N(1 ≦ i ≦ N-1)
The given graph is a tree.
InputThe input is given from Standard Input in the following format:
N
a_1 b_1
:
a_{N-1} b_{N-1}
OutputPrint the minimum k such that the given tree is a tree of uninity k.
Sample Input 17
1 2
2 3
2 4
4 6
6 7
7 5
Sample Output 12
A tree of uninity 1 consisting of vertices 1, 2, 3 and 4 can be constructed from the following: a tree of uninity 0 consisting of vertex 1, a tree of uninity 0 consisting of vertex 3, a tree of uninity 0 consisting of vertex 4, and vertex 2.
A tree of uninity 1 consisting of vertices 5 and 7 can be constructed from the following: a tree of uninity 1 consisting of vertex 5, and vertex 7.
A tree of uninity 2 consisting of vertices 1, 2, 3, 4, 5, 6 and 7 can be constructed from the following: a tree of uninity 1 consisting of vertex 1, 2, 3 and 4, a tree of uninity 1 consisting of vertex 5 and 7, and vertex 6.
Sample Input 212
1 2
2 3
2 4
4 5
5 6
6 7
7 8
5 9
9 10
10 11
11 12
Sample Output 23
Extracted Specification
An integer N (2 ≤ N ≤ 10^5), representing some quantity or size.
N-1 pairs of integers (a_i, b_i) where each integer is between 1 and N.
### Example Input:
```
7
1 2
2 3
2 4
4 6
6 7
7 5
```
### Function Signature:
Write a function f(N, edges) that takes in the input.
def f(N: int, edges: List[Tuple[int, int]]):
'''
N: an integer
edges: a list of tuples of integers
'''
Test Case Generator
Sure, I can help you with that. The problem seems to be about finding the minimum "uninity" of a given tree. Let's create a test case generator for this problem.
Here's a Python code snippet for a test case generator:
```python
import random
def generate_tree_edges(N):
# Generate a random tree with N nodes
edges = []
for i in range(2, N + 1):
# Connect each node i with a random node from 1 to i-1 to ensure it's a tree
edges.append((i, random.randint(1, i - 1)))
return edges
def tcgen():
# Randomly choose N within the constraints
N = random.randint(2, 10**5)
# Generate the tree edges
edges = generate_tree_edges(N)
# Format the input as a string
input_data = f"{N}\n" + "\n".join(f"{a} {b}" for a, b in edges)
return input_data
# Example usage:
print(tcgen())
```
### Explanation of the Code
1. **`generate_tree_edges(N)` Function**:
- This function generates `N-1` edges to form a tree with `N` nodes.
- It ensures that the generated edges form a tree by connecting each node `i` to a random node in the range `[1, i-1]`.
2. **`tcgen()` Function**:
- It randomly selects a value for `N` within the given constraints (`2 ≤ N ≤ 10^5`).
- It generates the tree edges by calling `generate_tree_edges(N)`.
- It formats the edges into a string that matches the input format for the problem.
### Example Output
The generated test cases will look something like this:
```
5
2 1
3 2
4 3
5 1
```
Each time you run `tcgen()`, it will produce a different valid test case for the problem, ensuring a variety of scenarios for testing your solution.
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