Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.
**
   For large \(x\), we can write:
   \[
   f(x) = ax + b
   \]
   where \(b\) is a constant. To determine \(b\), consider the original condition for \(x < c\) and \(y\) very large. The quotient must be 1, so:
   \[
   f(x) = ax + 1
   \]
   for \(x < c\).

7. **Conclusion:**
   For \(x, y > c\) and very large, we have \(b = 1\). Thus, the general form of the solution is:
   \[
   f(x) = ax + 1
   \]
   where \(a\) is a positive integer. These solutions satisfy the given condition.

The final answer is \( \boxed{ f(x) = ax + 1 } \) where \(a\) is a positive integer.