There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

1. **Prove that we cannot cut more than 2500 trees:**

   - Consider the square grid of trees with coordinates ranging from \((0,0)\) to \((99,99)\).
   - Divide this grid into \(2500\) smaller squares, each of size \(2 \times 2\). Each smaller square contains four points: \((2i, 2j)\), \((2i+1, 2j)\), \((2i, 2j+1)\), and \((2i+1, 2j+1)\) for \(0 \leq i, j \leq 49\).
   - Within each \(2 \times 2\) square, any tree stump can see the other three tree stumps. Therefore, at most one tree in each \(2 \times 2\) square can be cut down to ensure that no two stumps can see each other.
   - Since there are \(2500\) such \(2 \times 2\) squares, the maximum number of trees that can be cut down is \(2500\).

2. **Prove that we can cut down 2500 trees:**

   - We will show that it is possible to cut down all trees with even-numbered coordinates without violating the condition.
   - Consider the trees at coordinates \((2i, 2j)\) for \(0 \leq i, j \leq 49\). These are the bottom-left trees in each \(2 \times 2\) square.
   - We need to prove that no two stumps at these coordinates can see each other. Consider two such points \(P = (2a, 2b)\) and \(Q = (2c, 2d)\) where \(a, b, c, d \in \{0, 1, \ldots, 49\}\) and \((a, b) \neq (c, d)\).
   - The line segment \(PQ\) must contain at least one lattice point with at least one odd coordinate to ensure that no two stumps can see each other.
   - The midpoint of \(PQ\) is \(\left(\frac{2a + 2c}{2}, \frac{2b + 2d}{2}\right) = (a + c, b + d)\). Since \(a \neq c\) or \(b \neq d\), at least one of \(a + c\) or \(b + d\) is odd, ensuring that the midpoint is not a point with even coordinates.
   - Therefore, the segment \(PQ\) contains a point with at least one odd coordinate, ensuring that no two stumps can see each other.

Thus, it is possible to cut down exactly \(2500\) trees such that no two stumps can see each other.

The final answer is \(\boxed{2500}\)