Non-degeneracy conditions for plane branches

Let $x=t^n$, $y=\sum_{i=1}^{\infty}a_it^i$ be a parametrisation of the germ of a complex plane analytic curve $\Gamma$ at the origin. Then $\Gamma$ has the implicit equation $f(x,y)=0$ in the neighbourhood of the origin, where $f=\sum c_{ij}x^iy^j$ is a Weierstrass polynomial in $\mathbb{C}[[x]][y]$ of degree $n$. Every polynomial depending on coefficients $c_{ij}$ can be expressed as a polynomial depending on the coefficients $a_i$.