I have the following branching process:
(I added the $Z$'s myself for the number of individuals in each gen, hope it's correct.)
Assume the offspring distribution is Poisson with expectation $\lambda$ and also assume that we use an improper prior $\pi(\lambda)\propto_{\lambda}1/\lambda$ for $\lambda$. Using the information in the figure, compute the posterior distribution for $\lambda.$
Using Baye's theorem with an observed count $k$, I get
$$\pi(\lambda|k)\propto_{\lambda}\pi(k|\lambda)\pi(\lambda)\propto_{\lambda}e^{-\lambda}\cdot\lambda^{k}\cdot\frac{1}{\lambda}=e^{-\lambda}\lambda^{k-1}.$$
So we see that this posterior is $\pi(\lambda|k)\sim\Gamma(\lambda;k,1).$
However, how do I know what $k$ is based on the figure?