  If $n \geq 3$, then moduli space ${\mathcal M}_{0,[n+1]}$, of isomorphisms
classes of $(n+1)$-marked spheres, is a complex orbifold of dimension $n-2$.
Its branch locus ${\mathcal B}_{0,[n+1]}$ consists of the isomorphism classes
of those $(n+1)$-marked spheres with non-trivial group of conformal
automorphisms. We prove that ${\mathcal B}_{0,[n+1]}$ is connected if either $n
\geq 4$ is even or if $n \geq 6$ is divisible by $3$, and that it has exactly
two connected components otherwise. The orbifold ${\mathcal M}_{0,[n+1]}$ also
admits a natural real structure, this being induced by the complex conjugation
on the Riemann sphere. The locus ${\mathcal M}_{0,[n+1]}({\mathbb R})$ of its
fixed points, the real points, consists of the isomorphism classes of those
marked spheres admitting an anticonformal automorphism. Inside this locus is
the real locus ${\mathcal M}_{0,[n+1]}^{\mathbb R}$, consisting of those
classes of marked spheres admitting an anticonformal involution. We prove that
${\mathcal M}_{0,[n+1]}^{\mathbb R}$ is connected for $n \geq 5$ odd, and that
it is disconnected for $n=2r$ with $r \geq 5$ is odd.
