Summary: In this paper we present a construction of every $k$-ary tree using a forest of $(k - 1)$-ary trees satisfying a particular condition. We use this method recursively for the construction of the set of $k$-ary trees from the set of $(k - 1)$-Dyck paths, thus obtaining a new bijection $\phi $ between these two sets. Furthermore, we introduce a new order on $[k]^{*}$ which is used for the full description of this bijection. Finally, we study some new statistics on $k$-ary trees which are transferred by $\phi $ to statistics concerning the occurrence of strings in $(k - 1)$-Dyck paths.

05A15, 05A19

generalized Dyck words, k-ary trees, k-Catalan numbers