Certainly! Here is a Python function to generate test cases for the problem "G: Treasure Hunter":

```python
import random

def generate_test_case():
    # Constraints
    N = random.randint(1, 100)  # Number of piles
    W = random.randint(1, 100000)  # Budget for road construction
    
    # Generating treasures in each pile
    p = [random.randint(1, 1000000000) for _ in range(N)]
    
    # Generating road connections and their costs
    roads = []
    for _ in range(N - 1):
        u = random.randint(1, N - 1)
        v = random.randint(u + 1, N)
        c = random.randint(1, 100000)
        roads.append((u, v, c))
    
    # Preparing the input format
    input_data = f"{N} {W}\n"
    input_data += " ".join(map(str, p)) + "\n"
    for u, v, c in roads:
        input_data += f"{u} {v} {c}\n"
    
    return input_data

# Example usage
if __name__ == "__main__":
    print(generate_test_case())
```

This function generates a random test case for the given problem, adhering to the constraints provided. The `N` piles and the budget `W` are chosen randomly within their respective ranges. The treasures in each pile and the roads between piles with their construction costs are also generated randomly. The function then formats this data as a string suitable for use as input for the problem.

You can run this function to generate different test cases each time.