Recursion, Catalan Numbers & Generating Functions · Superchallenge problem

Let $C(x)=\sum_{n\ge 0}C_nx^n$ satisfy $C(x)=1+xC(x)^2$. Solve for $C(x)$, then use the binomial series to recover $C_n=\frac{1}{n+1}\binom{2n}{n}$; state the value of $C_5$.