Showing asymptotic lower bound on log of recurrence
Your sequence $T(n)$ is, in fact, the Catalan numbers, as show on the On-Line Encyclopedia of Integer Sequences. You can also take a look at the Wikipedia article on Catalan numbers, which shows its many applications in combinatorics.
$$T(n)=C(n-1)=\frac{(2(n-1))!}{n!(n-1)!}\sim \frac{4^{n-1}}{n^{3/2}\sqrt{\pi}}.$$
In particular,
$$\log T(n) = (n-1)\log 4 - \frac32\log n - \frac12\log\pi + o(1),$$
or,$$\log T(n) \sim n\log 4.$$