Despite the wealth of Planck results, there are difficulties in disentangling the primordial non-Gaussianity of the Cosmic Microwave Background (CMB) from the secondary and the foreground non-Gaussianity (NG). For each of these forms of NG the lack of complete data introduces model-dependences. Aiming at detecting the NGs of the CMB temperature anisotropy $\delta T$ , while paying particular attention to a model-independent quantification of NGs, our analysis is based upon statistical and morphological univariate descriptors, respectively: the probability density function $P(\delta T)$ , related to v(0), the first Minkowski Functional (MF), and the two other MFs, v(1) and v(2). From their analytical Gaussian predictions we build the discrepancy functions ${{ \Delta }_{k}}$ (k  =  P, 0, 1, 2) which are applied to an ensemble of 10(5) CMB realization maps of the $ \Lambda $ CDM model and to the Planck CMB maps. In our analysis we use general Hermite expansions of the ${{ \Delta }_{k}}$ up to the 12th order, where the coefficients are explicitly given in terms of cumulants. Assuming hierarchical ordering of the cumulants, we obtain the perturbative expansions generalizing the second order expansions of Matsubara to arbitrary order in the standard deviation ${{\sigma}_{0}}$ for $P(\delta T)$ and v(0), where the perturbative expansion coefficients are explicitly given in terms of complete Bell polynomials. The comparison of the Hermite expansions and the perturbative expansions is performed for the $ \Lambda $ CDM map sample and the Planck data. We confirm the weak level of non-Gaussianity (1–2)σ of the foreground corrected masked Planck 2015 maps.