k-non-crossing trees and edge statistics modulo k

Instead of $k$-Dyck paths we consider the equivalent concept of $k$-non-crossing trees. This is our preferred approach relative to down-step statistics modulo $k$ (first studied by Heuberger, Selkirk, and Wagner by different methods). One symmetry argument about subtrees is needed and the rest goes along the lines of a paper by Flajolet and Noy.