We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.
1. **Understanding the Problem:**
   We have a $100 \times 100$ garden, which means there are $10000$ individual $1 \times 1$ squares, each containing exactly one tree. We need to find the maximum number of trees that can be cut such that between any two cut trees, there is at least one uncut tree.

2. **Lemma:**
   In any $2 \times 2$ square, we can have at most one cut tree. If there were more than one cut tree in a $2 \times 2$ square, there would be a pair of cut trees with no uncut tree between them, violating the problem's condition.

3. **Reformulating the Garden:**
   Consider the $100 \times 100$ garden as a $50 \times 50$ grid of $2 \times 2$ squares. Each $2 \times 2$ square can have at most one cut tree according to the lemma.

4. **Counting the Maximum Number of Cut Trees:**
   Since there are $50 \times 50 = 2500$ such $2 \times 2$ squares in the $100 \times 100$ garden, and each can have at most one cut tree, the maximum number of cut trees is $2500$.

5. **Constructing an Arrangement:**
   To achieve this maximum, we can cut the tree in the lower left square of each $2 \times 2$ square. This ensures that no two cut trees are adjacent within any $2 \times 2$ square, satisfying the condition that between any two cut trees, there is at least one uncut tree.

Thus, the maximum number of trees that can be cut is $2500$.

The final answer is $\boxed{2500}$.