How to calculate the time complexity of a Catalan-like recurrence by substitution?
You can use induction. First, check directly that $P(n) \geq 2^{n-2}$ for all $n \leq 4$ (the relevant values are $1,1,2,5$). Now suppose that $P(n) \geq 2^{n-2}$ holds for all $n \leq m$, where $m \geq 4$. Then
$$
P(m+1) = \sum_{k=1}^m P(k) P(m+1-k) \geq \sum_{k=1}^m 2^{k-2} 2^{m+1-k-2} \geq m 2^{m+1-4} \geq 2^{m+1-2},
$$
since $m \geq 4$.