  The investigation of the graph $\mathcal{G}_p$ associated with the finite
$p$-groups of maximal class was initiated by Blackburn (1958) and became a deep
and interesting research topic since then. Leedham-Green and McKay (1976-1984)
introduced skeletons of $\mathcal{G}_p$, described their importance for the
structural investigation of $\mathcal{G}_p$ and exhibited their relation to
algebraic number theory. Here we go one step further: we partition the
skeletons into so-called Galois trees and study their general shape. In the
special case $p \geq 7$ and $p \equiv 5 \bmod 6$, we show that they have a
significant impact on the periodic patterns of $\mathcal{G}_p$ conjectured by
Eick, Leedham-Green, Newman and O'Brien (2013). In particular, we use Galois
trees to prove a conjecture by Dietrich (2010) on these periodic patterns.
