We shall present and discuss the notion of essential paths on graphs (Ocneanu) and use this notion to define the (first) algebra structure on the quantum groupoid associated with a graph G (here, we shall take G to be a tree). This algebra is finite dimenional when G is of type ADE. We shall also discuss the relations between essential paths, representation theory of SU2, Temperlie - Lieb - Jones algebras and with the Gelfand - Ponomarev construction for quivers. Genralizations to SLn will also be discussed.