Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis

We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of "petals" in all the flowers (petals' weight) and the number of edges in the petals of all the flowers (flowers' weight). We give explicit expressions of their generating functions and deduce general formulas for the asymptotic growth of their coefficients and the expectations of their concentrated distributions.