Studying the volume of moduli space of Riemann surfaces is a central problem in string theory. In particular, the computation of Gromov-Witten invariants of Toric Calabi-Yau threefolds is a fascinating subject where many different aspects of physics and mathematics beautifully coexist. Following methods developed in the context of random matrix theory, it has been recently conjectured that this problem can be solved simply by induction on the Euler characteristic of the surfaces enumerated. In this talk, I will briefly present this conjecture and explain its proof in the simplest cases by showing the equivalence between this recursion and the topological vertex formalism which is known to compute generating functions of these invariants. For this purpose, I will explain how the main character of this topological recursion, the spectral curve, arises from mirror symmetry. I will then show the use of promoting the considered generating functions to differential forms on the spectral curve. Indeed, expanding these forms at different points of the spectral curve allows to go from the topological vertex formalism to the topological recursion.