Arithmetic decompositions of natural numbers

Every natural number can be additively partitioned so for example 3 can be decomposed to 1+1+1 or 1+2 and every number can be multiplicatively partitioned so for example 4 can be decomposed into 2*2. Arithmetic decompositions combine both additive and multiplicative partitioning:

0,1,(+ 1 1), 2,  (+ 1 1 1), (+ 1 2), 3, 
(* (+ 1 1) (+ 1 1)), (* 2 (+ 1 1)), (* 2 2), 
(+ 1 1 1 1), (+ 1 1 2), (+ 2 2), (+ 1 3), 4, 
(+ 1 (* (+ 1 1) (+ 1 1)), (+ 1 (* 2 (+ 1 1))), 
(+ 1 (* 2 2)), (+ 1 1 1 1 1), (+ 1 1 1 2), 
(+ 1 2 2), (+ 1 1 3), (+ 2 3), 5

I'm not familiar with any integer sequence that enumerates the arithmetic decompositions nor any applications of their definition, however, I think it is interesting to consider how they might fit into a generalized decomposition system.