A (p, q)-graph G is called super edge-magic if there exists a bijective function f : V (G) ∪E(G) → {1, 2, . . . , p+q} such that f(u)+f(v)+f(uv) is a constant for each uv ∈ E(G) and f(V (G)) ={1, 2, . . . , p}.In this paper, we introduce the concept of strong super edge-magic labeling as a particular classof super edge-magic labelings and we use such labelings in order to show that the number of superedge-magic labelings of an odd union of path-like trees (mT), all of them of the same order, grows atleast exponentially with m.