曾江教授重新发现了Ramanujan所研究的多项式与Dumont教授给出的有根标号树的非恰当边之间的联系，后来该多项式被陈永川教授称为Ramanujan-Shor多项式；Gessel教授与Seo教授讨论了Gould多项式与有根树的根度数与非恰当点个数的联合分布的联系。这个联系也给出了有根树上根度数、恰当点和非恰当点的对称性。γ-正性是组合多项式的非常重要的性质，关于Cayley树生成函数的γ-正性的研究则刚刚开始起步。.本项目中，我们将基于之前给出的刻画Ramanujan-Shor多项式的上下文无关文法，研究树上相关多项式的性质，具体而言，我们将尝试给出一个细化的文法，同时刻画Gould多项式和Ramanujan-Shor多项式，并基于此解决这两个多项式相关的公开问题。此外，我们还将尝试使用文法研究排列和Cayley树上的γ-正性结论的统一框架。

J. Zeng established the connection between a polynomial studied by Ramanujan and the distribution of the number of improper edges of Cayley trees. And the polynomial is called Ramanujan-Shor polynomial by Chen. Gessel and Seo found that Gould polynomials can be seen as the generation polynomials of Cayley trees on the joint distribution of the degree of the root and the number of improper vertices, which also implies the symmetry of the number of proper vertices and the number of improper vertices of Cayley trees. The γ-positivity is an extremely important property of combinatorial polynomials. The γ-positivity of tee functions get less attention from combinatorist. .In this project, based on the context-free grammars associated with Ramanujan-Shor polynomials, we shall foucs on the properties of tree functions, such as Ramanujan-Shor polynomials, Gould polynomials and so on. Especially, we will provide a refined grammar to describe both Ramanujan-Shor polynonimals and Gould polynomials, and give the soultions for some open problems of these two polynomials. Finally, we shall establish a unified framework for studying the γ-positivity of descent polynomials of permutations and tree funcitons.