Increasing Binary Trees and the $(\alpha ,\beta )$-Eulerian Polynomials

In light of the grammar given by Ji for the $(\alpha ,\beta )$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the $\gamma$-coefficients of the $\alpha$-Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the $\gamma$-coefficients of the derangement polynomials in the same vein. By means of a decomposition of an increasing binary tree into a forest, we find combinatorial interpretations of the sums involving two identities of Ji, one of which can be viewed as $(\alpha ,\beta )$-extensions of the formulas of Petersen and Stembridge.