  The stratum $\mathcal{H}(a,-b_{1},\dots,-b_{p})$ of meromorphic $1$-forms
with a zero of order $a$ and poles of orders $b_{1},\dots,b_{p}$ on the Riemann
sphere has a map, the isoresidual fibration, defined by assigning to any
differential its residues at the poles. We show that above the complement of a
hyperplane arrangement, the resonance arrangement, the isoresidual fibration is
an unramified cover of degree $\frac{a!}{(a+2-p)!}$. Moreover, the monodromy of
the fibration is computed for strata with at most three poles and a system of
generators and relations is given for all strata. These results are obtained by
associating to special differentials of the strata a tree, and by studying the
relationship between the geometric properties of the differentials and the
combinatorial properties of these trees.
