  This article explores to which extent the algebro-geometric theory of
rational descendant Gromov-Witten invariants can be carried over to the
tropical world. Despite the fact that the tropical moduli-spaces we work with
are non-compact, the answer is surprisingly positive. We discuss the string,
divisor and dilaton equations, we prove a splitting lemma describing the
intersection with a "boundary" divisor and we prove general tropical versions
of the WDVV resp. topological recursion equations (under some assumptions). As
a direct application, we prove that the toric varieties $\mathbb{P}^1$,
$\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ and with Psi-conditions only
in combination with point conditions, the tropical and classical descendant
Gromov-Witten invariants coincide (which extends the result for $\mathbb{P}^2$
in Markwig-Rau-2008). Our approach uses tropical intersection theory and can
unify and simplify some parts of the existing tropical enumerative geometry
(for rational curves).
