  Ma-Ma-Yeh made a beautiful observation that a transformation of the grammar
of Dumont instantly leads to the $\gamma$-positivity of the Eulerian
polynomials. We notice that the transformed grammar bears a striking
resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The
appearance of the factor of two fits perfectly in a grammatical labeling of
0-1-2 increasing plane trees. Furthermore, the grammatical calculus is
instrumental to the computation of the generating functions. This approach can
be adapted to study the $e$-positivity of the trivariate second-order Eulerian
polynomials first introduced by Dumont in the contexts of ternary trees and
Stirling permutations, and independently defined by Janson, in connection with
the joint distribution of the numbers of ascents, descents and plateaux over
Stirling permutations.
