Consider the branching process with offspring distribution given by \( P(X=0)=\alpha\), \( P(X=1)=\frac23-\alpha\) and \( P(X=2)=\frac13\) for some \(\alpha \in [0,\frac23]\).

1. For what values of \(\alpha\) is the process certain to die out. ?
2. For values where there is a probability of surviving forever, what is this probability as a function of \(\alpha\) ?