  We consider two seemingly unrelated problems, the calculation of the WKB
expansion of the harmonic oscillator wave functions and the counting the number
of Feynman diagrams in QED or in many-body physics and show that their
solutions are both encoded in a single enumerative problem, the calculation of
the number of certain types of ribbon graphs. In turn, the numbers of such
ribbon graphs as a function of the number of their vertices and edges can be
determined recursively through the application of the topological recursion of
Eynard-Orantin to the algebraic curve encoded in the Schr\"odinger equation of
the harmonic oscillator. We show how the numbers of these ribbon graphs can be
written down in closed form for any given number of vertices and edges. We use
these numbers to obtain a formula for the number of N-rooted ribbon graphs with
e edges, which is the same as the number of Feynman diagrams for 2N-point
function with e+1-N loops.
