On the representation of simply generated trees by leftist trees

Each simply generated family F of trees is unambiguously associated with another simply generated family F1 of trees such that the total weight of the trees with m leaves in F is equal to the total weight of the leftist trees with m leaves in F1. This interrelation induces a partition P on the set of trees with m leaves appearing in F such that each block B∈P is associated with exactly one leftist tree τB∈F1 with m leaves. Given τB (resp. T∈B), we shall establish an explicit transformation defined on the trees for the construction of B (resp. τB). This approach implies a compressed representation of a simply generated family of trees with m leaves by leftist simply generated trees with the same number of leaves.