Labeled Plane Trees and Increasing Plane Trees

This note is dedicated to presenting a polynomial analogue of $(n+1)!C_n=2^n(2n-1)!!$ (with $C_n$ as the $n$-th Catalan number) in the context of labeled plane trees and increasing plane trees, based on the definition of improper edges in labeled plane trees. A new involution on labeled plane trees is constructed to establish this identity, implying that the number of improper edges and the number of proper edges are equidsitributed over the set of labeled plane trees.