  Goussarov-Habiro's theory of clasper surgeries defines a filtration of the
monoid of string links $L(m)$ on $m$ strands, in a way that geometrically
realizes the Feynman diagrams appearing in low-dimensional and quantum
topology. Concretely, $L(m)$ is filtered by $C_n$-equivalence, for $n\geq 1$,
which is defined via local moves that can be seen as higher crossing changes.
The graded object associated to the Goussarov-Habiro filtration is the
Goussarov-Habiro Lie algebra of string links $\mathcal{L} L(m)$. We give a
concrete presentation, in terms of primitive Feynman (tree) diagrams and
relations ($\text{1T}$, $\text{AS}$, $\text{IHX}$, $\text{STU}^2$), of the
rational Goussarov-Habiro Lie algebra $\mathcal{L} L(m)_{\mathbb{Q}}$. To that
end, we investigate cycles in graphs of forests: flip graphs associated to
forest diagrams and their $\text{STU}$ relations. As an application, we give an
alternative diagrammatic proof of Massuyeau's rational version of the
Goussarov-Habiro conjecture for string links, which relates
indistinguishability under finite type invariants of degree $<n$ and
$C_n$-equivalence.
