  This paper is motivated by two problems recently proposed by Coker on
combinatorial identities related to the Narayana polynomials and the Catalan
numbers. We find that a bijection of Chen, Deutsch and Elizalde can be used to
provide combinatorial interpretations of the identities of Coker when it is
applied to weighted plane trees. For the sake of presentation of our
combinatorial correspondences, we provide a description of the bijection of
Chen, Deutsch and Elizalde in a slightly different manner in the form of a
direct construction from plane trees to 2-Motzkin paths without the
intermediate step involving the Dyck paths.
