The number of plane oriented recursive trees is $(2n-3)!!$
I understand that given a vertex $v$ with $k$ successors, there are $k+1$ ways to attach a new vertex to create a new tree of size one bigger than before. So then if there are $n$ vertices in the tree there should be $2n-3$ ways to attach another vertex, but I don't follow this step.
My thinking was that for each vertex you have "number of succesors"+1 possibilities for attachment, i.e. $\sum_{v \in V}(d^+(v)+1) = 2n-1$, for V the vertices in the tree at the moment, $d^+(v)$ the number of successors of $v$ and $n =|V|$. It seems like I'm overcounting, though.
You're counting right – it's just that the last tree that you attach a new vertex to is the one with $n-1$ vertices, not the one with $n$ vertices – thus the last factor is $2(n-1)-1=2n-3$. See also this treatment.