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.