Let c_n be the number of objects, such as phylogenetic networks, of size n. We are interested in the asymptotics of this sequence, i.e., a "simpler" sequence a_n such that the quotient c_n/a_n converges to 1 (or is bounded) for n to infinity. In particular, we will focus on stretched exponential term m^(n^(s)) with m>0 and 0<s<1. The presence of such a term is not common, although recently more and more examples emerge. It is generally quite difficult to prove that a sequence has such a stretched exponential, which is partly due to the observation that such a sequence cannot be "very nice", as, e.g., its generating function cannot be algebraic. Previously, the saddle point method was the only generic method for proving such a phenomenon, which, however, requires detailed information on the generating function. Recently, together with Andrew Elvey Price and Wenjie Fang, we have developed a new method on the level of recurrences to prove stretched exponentials. I will introduce the basics of this method and show how we used it together with Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, and Guan-Ru Yu to prove such a phenomenon for d-combining tree-child networks, i.e. networks in which every reticulation node has exactly d parents.