  We will describe the topological type of the discriminant curve of the
morphism $(\ell, f)$, where $\ell$ is a smooth curve and $f$ is an irreducible
curve (branch) of multiplicity less than five or a branch that the difference
between its Milnor number and Tjurina number is less than 3. We prove that for
a branch of these families, the topological type of the discriminant curve is
determined by the semigroup, the Zariski invariant and at most two other
analytical invariants of the branch.
