Total Vertex Enumeration in Rooted Planar Maps

Two combinatorial identities are proved:
(1) \(\quad H_n(\varepsilon) = \frac{n+2}{3} M_n(\varepsilon)\), where \(H_n(\varepsilon)\) denotes the total number of vertices in all the n-edged rooted planar Eulerian maps and \(M_n(\varepsilon)\) denotes the number of such maps.
(2) \(\quad H_n(\mathcal{L}) = \frac{5n^2+13n+2}{2(4n+1)} M_{n }(\mathcal{L})\), where \(H_n(\mathcal{L})\) and \(M_{n}(\mathcal{L})\) are defined similarly for the class \(\mathcal{L}\) of loopless maps.
Simple closed formulae for \(M_n(\varepsilon)\) and \(M_{n}(\mathcal{L})\) are well known, and they correspond to equivalent binomial sum identities.