  By considering the parity of the degrees and levels of nodes in increasing
trees, a new combinatorial interpretation for the coefficients of the Taylor
expansions of the Jacobi elliptic functions is found. As one application of
this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved.
Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou
introduced weakly increasing trees on a multiset. A symmetry joint distribution
of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly
increasing trees is found, extending the Schett polynomials, a generalization
of the Jacobi elliptic functions introduced by Schett, to multisets. A
combinatorial proof and an algebraic proof of this symmetry are provided, as
well as several relevant interesting consequences. Moreover, via introducing a
group action on trees, we prove the partial $\gamma$-positivity of the multiset
Schett polynomials, a result implies both the symmetry and the unimodality of
these polynomials.
