Survival probability up to time $n$ in a branching process. Let $\{Z_n : n=0,1,2,\ldots\}$ be a Galton-Watson branching process with time-homogeneous offspring distribution
$$\mathbb P(Z_{n,j} = 0) = 1-p = 1 - \mathbb P(Z_{n,j}=2), $$
where $0<p<1$. That is, $Z_0 = 1$ and
$$Z_n = \sum_{j=1}^{Z_{n-1}}Z_{n,j}$$
for $n\geqslant 1$. Let 
$$T=\inf\{n : Z_n=0\} $$
be the extinction time of the process. I want to find
$$t_n := \mathbb P(T>n), $$
i.e. the probability that the process survives up to time $n$. I found the following recurrence:
$$t_{n+1} = p(2t_n - t_n^2)$$
(with $t_0=1$) from this mathoverflow question: https://mathoverflow.net/questions/87199/branching-process-survival-probability
I checked the recurrence for small values of $n$, and confirmed that $t:=\lim_{n\to\infty} t_n$ satisfies
$$ t = p(2t - t^2),$$
both in the case where $t=0$ $\left(p\leqslant\frac12\right)$ and where $t=2-\frac1p$ $\left(p>\frac12\right)$. So I'm fairly confident this recurrence is valid. However, I have no idea how to solve it.
For context, this problem comes from Adventures in Stochastic Processes by Sidney Resnick:

From @Did's comment, it appears to be intractable to find a closed form for $\mathbb P(T>n)$. I find it curious that the question would be asked were it not, though.

For p approaching 1, there is a solution, and I'll post a proof later, at least this will get the ball rolling.
For p=1 the recurrence relation solves to $$t_n =1-e^{c*2^{n}}$$
For p=2, certainly not applicable to this case, the recurrence relation solves to $$t_n=1-\cos(c2^n)$$
Proof
The original recurrence relation is given by
$$t_{n+1}=p(2t_n-{t_n}^2)$$
I'll prove the first solution. For $p=1$ assume that the first equation is a solution for a number n. Substitute the solution into the relation.
$$1-e^{2c2^n}=2-2e^{c2^n}-(1-e^{c2^n})^2$$
Expand and simplify
$$1-e^{2c2^n}=2-2e^{c2^n}-1+2e^{c2^n}-e^{2c2^n}=1-e^{2c2^n}$$
The second solution is proved in much the same way, you'd just need to know some trig identities.
Suggestions
Use a computer for most of the calculations and do interpolation between the points. If you feel adventurous, find a derivation for the $p=1$ solution, and perform perturbation analysis, you may or may not get a convergent solution. I'll repost if I get anything interesting.
