  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 Jacobi elliptic functions to
multisets. A combinatorial proof and an algebraic proof of this symmetry are
provided, as well as several relevant interesting consequences.
