  For a formation $\mathfrak{F}$ of finite groups, tight connections are
established between the pro-$\mathfrak{F}$-topology of a finitely generated
free group $F$ and the geometry of the Cayley graph
$\Gamma(\hat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion
$\hat{F_{\mathfrak {F}}}$ of $F$. For example, the Ribes--Zalesskii-Theorem is
proved for the pro-$\mathfrak{F}$-topology of $F$ in case
$\Gamma(\hat{F_{\mathfrak F}})$ is a tree-like graph. All these results are
established by purely geometric proofs, without the use of inverse monoids
which were indispensable in earlier papers, thereby giving more direct and more
transparent proofs. Due to the richer structure provided by formations
(compared to varieties), new examples of (relatively free) profinite groups
with tree-like Cayley graphs are constructed. Thus, new topologies on $F$ are
found for which the Ribes-Zalesskii-Theorem holds.
