= Solution
The <Monodromy theorem> states that if a holomorphic germ can be analytically continued along every path in a domain, its continuations along paths homotopic with fixed endpoints coincide. In a <simply connected domain> this gives a single-valued holomorphic function on the whole domain.
Write $\zeta_j=e^{2\pi ij/n}$. Remove the radial rays $R_j=\{t\zeta_j:t\geq1\}$ and set $U=\mathbb C\setminus\bigcup_jR_j$. This is an open star-shaped domain containing zero, hence simply connected, and $h=z^n-1$ never vanishes there. Locally its logarithm continues along all paths in $U$, so monodromy gives a holomorphic logarithm $L$ and
$$
\boxed{h^{1/r}=e^{L/r}\quad\text{on }U.}
$$
Equivalently, integrate $h'/h$ on the simply connected domain and choose the additive constant to obtain $e^L=h$.
Take $r$ copies of the slit domain, labeled by $j\in\mathbb Z/r\mathbb Z$, carrying values $e^{(L+2\pi ij)/r}$. Across each cut identify banks so that a positive loop around its simple zero increments the sheet index by one. This cyclic gluing produces the unbranched <Riemann surface> over $\mathbb C\setminus Z$, on which $w^r=z^n-1$ is single valued. Completing each finite puncture gives local coordinate $w$, with
$$
z-\zeta_j=\frac{w^r}{n\zeta_j^{n-1}}+O(w^{2r}).
$$
There is one ramification point of index $r$ over each of the $n$ roots of unity.
At infinity a loop around all $n$ zeros acts on sheet labels by $j\mapsto j+n\pmod r$. With $d=\gcd(n,r)$ this permutation has $d$ cycles, each of length $r/d$. In the compactification there are therefore $d$ points over infinity, each with ramification index $r/d$. Infinity is unbranched precisely when $d=r$, namely when $r\mid n$. Counting branch values in the base sphere,
$$
\boxed{\#\{\text{branch values of }\widetilde\pi\}=n\iff r\mid n;\quad\text{otherwise it is }n+1.}
$$
If “branch points” denotes ramification points upstairs instead, the count is $n$ or $n+d$ respectively, with exactly the same stated equivalence.
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