  We study constructible invariants of the moduli space
$\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves
to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles
specified by ${\boldsymbol{x}\in \mathbb{Z}^n}$. These spaces are central to
the enumerative geometry of $\mathbb{P}^1$, and provide a large family of
birational models of the Deligne--Mumford--Knudsen moduli space
$\overline{\mathcal{M}}_{0,n}$. For the sequence of vectors $\boldsymbol{x}$
corresponding to maps which are maximally ramified over $0$ and unramified over
$\infty$, we prove that a generating function for the topological Euler
characteristics of these spaces satisfies a differential equation which allows
for its recursive calculation. We also show that the class of the moduli space
in the Grothendieck ring of varieties is constant as $\boldsymbol{x}$ varies
within a fixed chamber in the resonance decomposition of $\mathbb{Z}^n$. We
conclude by suggesting several further directions in the study of these spaces,
giving conjectures on (1) the asymptotic behavior of the Euler characteristic
and (2) a potential chamber structure for the Chern numbers.
