To generate test cases for the given problem statement involving a tree with vertices and stones, we will need to create a function that generates a random tree structure, assigns a random number of stones to each vertex, and ensures the constraints are met.

Here is a Python function that does this:

```python
import random

def generate_tree_edges(N):
    """ Helper function to generate a list of edges forming a tree with N nodes """
    edges = []
    for i in range(2, N+1):
        edges.append((i, random.randint(1, i-1)))
    random.shuffle(edges)
    return edges

def tcgen():
    N = random.randint(2, 10**5)
    
    # Generate random stone counts for each vertex
    A = [random.randint(0, 10**9) for _ in range(N)]
    
    # Generate a random tree with N vertices
    edges = generate_tree_edges(N)
    
    return N, A, edges

# Example usage of the test case generator
if __name__ == "__main__":
    N, A, edges = tcgen()
    print(N)
    print(' '.join(map(str, A)))
    for edge in edges:
        print(f"{edge[0]} {edge[1]}")
```

### Explanation:

1. **`generate_tree_edges(N)`:**
   - This helper function generates a list of edges that form a tree with `N` nodes.
   - It ensures that the tree is connected by always linking node `i` to a random node from `1` to `i-1`. This guarantees that the graph is a valid tree.

2. **`tcgen()`:**
   - Generates a random integer `N` within the specified range.
   - Creates a list `A` of size `N`, where each element is a random integer between `0` and `10^9` representing the number of stones at each vertex.
   - Uses the `generate_tree_edges(N)` function to generate the list of edges, ensuring the resulting structure is a tree.
   - Returns `N`, `A`, and `edges` which represent the vertices, stone counts, and edges of the tree respectively.

3. **Example Usage:**
   - When `tcgen()` is called, it prints the formatted output similar to how the input would be provided for the problem:
     - The number of vertices `N`.
     - The list of stone counts `A`.
     - The edges of the tree.

This test case generator ensures that the generated inputs are within the constraints and form a valid tree structure, suitable for testing the problem solution.