  Consider a plane branch, that is, an irreducible germ of curve on a smooth
complex analytic surface. We define its blow-up complexity as the number of
blow-ups of points necessary to achieve its minimal embedded resolution. We
show that there are $F_{2n-4}$ topological types of blow-up complexity $n$,
where $F_{n}$ is the $n$-th Fibonacci number. We introduce
complexity-preserving operations on topological types which increase the
multiplicity and we deduce that the maximal multiplicity for a plane branch of
blow-up complexity $n$ is $F_n$. It is achieved by exactly two topological
types, one of them being distinguished as the only type which maximizes the
Milnor number. We show moreover that there exists a natural partial order
relation on the set of topological types of plane branches of blow-up
complexity $n$, making this set a distributive lattice, that is, any two of its
elements admit an infimum and a supremum, each one of these operations beeing
distributive relative to the second one. We prove that this lattice admits a
unique order-inverting bijection. As this bijection is involutive, it defines a
duality for topological types of plane branches. The type which maximizes the
Milnor number is also the maximal element of this lattice and its dual is the
unique type with minimal Milnor number. There are $F_{n-2}$ self-dual
topological types of blow-up complexity $n$. Our proofs are done by encoding
the topological types by the associated Enriques diagrams.
