  We describe a presentation for the descent algebra of the symmetric group
$\sym{n}$ as a quiver with relations. This presentation arises from a new
construction of the descent algebra as a homomorphic image of an algebra of
forests of binary trees which can be identified with a subspace of the free Lie
algebra. In this setting, we provide a new short proof of the known fact that
the quiver of the descent algebra of $\sym{n}$ is given by restricted partition
refinement. Moreover, we describe certain families of relations and conjecture
that for fixed $n\in\mathbb{N}$, the finite set of relations from these
families that are relevant for the descent algebra of $\sym{n}$ generates the
ideal of relations, and hence yields an explicit presentation by generators and
relations of the algebra.
