  We show that a coalescence equation exhibits a variety of critical behaviors,
depending on the initial condition. This equation was introduced a few years
ago to understand a toy model {studied by Derrida and Retaux to mimic} the
depinning transition in presence of disorder. It was shown recently that this
toy model exhibits the same critical behaviors as the equation studied in the
present work. Here we find several families of exact solutions of this
coalescence equation, in particular a family of scaling functions which are
closely related to the different possible critical behaviors. These scaling
functions lead to new conjectures, in particular on the shapes of the critical
trees, that we have checked numerically.
