  We propose a quantization algebra of the Loday-Ronco Hopf algebra
$k[Y^\infty]$, based on the Topological Recursion formula of Eynard and
Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of
planar binary trees is a space of solutions for the genus 0 version of
Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as
to include some new graphs with loops is the correct setting to find a solution
space for arbitrary genus. Here we show that this new algebra $k[Y^\infty]_h$
is still a Hopf algebra that can be seen in some sense to be made precise in
the text as a quantization of the Hopf algebra of planar binary trees, and that
the solution space of Topological Recursion $\mathcal{A}^h_{\text{TopRec}}$ is
a subalgebra of a quotient algebra $\mathcal{A}_{\text{Reg}}^h$ obtained from
$k[Y^\infty]_h$ that nevertheless doesn't inherit the Hopf algebra structure.
We end the paper with a discussion on the cohomology of
$\mathcal{A}^h_{\text{TopRec}}$ in low degree.
