1. **State the problem:** We have 17 squirrels sitting on 4 trees. Each tree has at least 2 squirrels. The number of squirrels on each tree is different. We want to find the largest possible number of squirrels on one tree. 2. **Define variables:** Let the number of squirrels on the four trees be $a, b, c, d$. We know: - $a + b + c + d = 17$ - $a, b, c, d \geq 2$ - All $a, b, c, d$ are distinct integers. 3. **Goal:** Maximize the largest number among $a, b, c, d$. 4. **Approach:** To maximize the largest number, minimize the other three while respecting constraints. 5. **Minimize the three smallest:** Since each tree has at least 2 squirrels and all numbers are distinct, the smallest three distinct integers each at least 2 are 2, 3, and 4. 6. **Calculate the sum of the three smallest:** $$2 + 3 + 4 = 9$$ 7. **Find the largest number:** $$d = 17 - 9 = 8$$ 8. **Check distinctness:** The numbers are 2, 3, 4, and 8, all distinct. 9. **Check minimums:** All are at least 2. 10. **Conclusion:** The largest possible number of squirrels on one tree is $\boxed{8}$.