  Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we
come to the realization that the grammatical labels play a role as records of
chopped off leaves of
  the original increasing binary trees. While such an understanding is purely
psychological, it does give rise to an efficient apparatus to tackle the
partial $\gamma$-positivity of the Eulearian polynomials on multiset Stirling
permutations, as long as we bear in mind the combinatorial meanings of the
labels $x$ and $y$ in the Gessel representation of a $k$-Stirling permutation
by means of an increasing $(k+1)$-ary tree. More precisely, we introduce a
Foata-Strehl action on the Gessel trees resulting in an interpretation of the
partial $\gamma$-coefficients of the aforementioned Eulerian polynomials,
different from the ones found by Lin-Ma-Zhang and Yan-Huang-Yang. In
particular, our strategy can be adapted to deal with the partial
$\gamma$-coefficients of the second order Eulerian polynomials, which in turn
can be readily converted to the combinatorial formulation due to Ma-Ma-Yeh in
connection with certain statistics of Stirling permutations.
