For great diagrams relating to the Catalan number and the number of monotonic paths not crossing the diagonal, see this: http://en.wikipedia.org/wiki/Catalan_number#Second_proof
So why is the number of monotonic paths not crossing the diagonal the Catalan number?
I understand that through the "reflection method", we arrive at $2n \choose n$ - $2n \choose n-1$ which is indeed the definition of the Catalan number.
However, a definition of the Catalan number is also given by the recurrence relation: $C_0 = 1 \quad \mbox{and} \quad C_{n+1}=\sum_{i=0}^{n}C_i\,C_{n-i}\quad\text{for }n\ge 0$
Can anyone point out a way to intuitively visualize the solution for the number of monotonic paths not crossing the diagonal in terms of this recurrence relation instead of the explicit formula of the "reflection method" above?