  In this note, by counting some colored plane trees we obtain several binomial
identities. These identities can be viewed as specific evaluations of certain
generalizations of the Narayana polynomials. As consequences, it provides
combinatorial proofs for a bijective problem in Stanley's collection "Bijective
Proof Problems", a new formula for the Narayana polynomials as well as a new
expression for the Harer-Zagier formula enumerating unicellular maps, in a
unified way. Furthermore, we identify a class of plane trees, whose enumeration
is closely connected to the Schr\"oder numbers. Many other binomial identities
are presented as well.
