As noted in the OEIS entry, the sequence is bounded below by the number of unlabelled rooted trees, so the radius of convergence can be only equal or less. For unlabelled rooted trees, the radius of convergence is the reciprocal of Otter's constant $\rho=2.955765285{\ldots}$ (see A051491 for lots of digits). This is suspiciously close to Roland's numerical estimate so maybe the radius of convergence is the same.
Wikipedia kindly provides both sequences up to $n=1000$, which allows a numerical comparison. Let $a_n$ be the counts in Roland's question and $b_n$ be the number of unlabelled rooted trees. For $100\le n\le 1000$, the ratio $a_n/b_n$ is a very close match to $$ 3.04 \times 1.00406^n.$$ This suggests that the radius of convergence of $\{a_n\}$ is a tiny bit less than $1/\rho$. Established methods for asymptotically counting trees are likely to give precise results.