In this paper we study a model of Schröder trees whose labelling is increasing along the branches. Such tree family is useful in the context of phylogenetic. The tree nodes are of arbitrary arity (i.e. out-degree) and the node labels can be repeated throughout different branches of the tree. Once a formal construction of the trees is formalized, we then turn to the enumeration of the trees inspired by a renormalisation due to Stanley on acyclic orientations of graphs. We thus exhibit links between our tree model and labelled graphs and prove a one-to-one correspondence between a subclass of our trees and labelled graphs. As a by-product we obtain a new natural combinatorial interpretation of Stanley’s renormalising factor. We then study different combinatorial characteristics of our tree model and finally, we design an efficient uniform random sampler for our tree model based on the classical recursive generation method.