  We first review our previous work arxiv:1503.02993 [math-ph] where we
considered a model for topological recursion based on the Hopf Algebra of
planar binary trees of Loday and Ronco and showed that extending this Hopf
Algebra by identifying pairs of nearest neighbor leaves and thus producing
graphs with loops we obtain the full recursion formula of Eynard and Orantin.
Then we discuss the algebraic structure of the spaces of correlation functions
in g = 0 and in g > 0. By taking a classical and a quantum product respectively
we endow both spaces with a ring structure. This is an extended version of the
contributed talk given at the 2016 von Neumann Symposium on Topological
Recursion and its Influence in Analysis, Geometry and Topology, from 4 to 8
July 2016 at Hilton Charlotte University Place, USA.
