How many unlabelled oriented trees with $n$ vertices?

Question

Is there a known upper bound of the form $a^n$ on the number of unlabelled oriented trees with $n$ vertices?

Answer 1

Yes: the Catalan numbers count unlabeled ordered trees with $n$ vertices (where an ordered tree is an oriented tree in which the order of the children of a node matters) and therefore the $n^{\text{th}}$ Catalan number $C_n$ is an upper bound on the number of unlabeled oriented trees. But $C_n = \frac1{n+1}\binom{2n}{n} \le 4^n$.