graph theory - Bound on the number of unlabeled cographs on n vertices

See e.g. the Wikipedia article on cographs, which explains that isomorphism classes of cographs are in one-to-one correspondence with isomorphism classes of n-leaf rooted trees in which the internal nodes are labeled with 0's and 1's and in which, moreover, the labels are in strict alternation from root to leaf.

Because of the alternation, the labeling part only adds a factor of two to the overall count, so really all you need to do is to count n-leaf rooted trees. So the number of cographs appear to be the numbers in OEIS sequence A000084 (the number of trees is half that, A000669). They are asymptotic to around 3.561^n, matching Zaimi's answer.