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 φ 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 φ to statistics concerning the occurrence of strings in (k - 1)-Dyck paths.