Galton Watson tree with various kinds of offspring

As far as I understood, for the Galton-Watson tree process, the offspring are of one type. I am thinking of the case where we have offspring of different types. I have illustrated this in the example below: It can be seen that we have both sad face and happy face, rather than just a single type emoji. I am interested to find out the expected height of such trees for a given face (happy or sad). Let us assume that the reproduction goes with Poisson distribution.

I know that if we had only one type of children we could write the expectation via a generating function. First the generating function would be: $$f(s)=\sum_{k=0}^{\infty}P(Z_n=k)s^k$$ where $$k$$ is the number of children and $$P(Z_n=k)$$ the probability and $$s$$ is a dummy variable in that case we can define: $$m=E[Z_n]=f'(s)|_{s=1}=\sum_{k=0}^{\infty}P(Z_n=k)k$$ I am not sure but I think for the two type (Happy and Sad) we should have something like: $$E[Z_0]=\sum_{n=0}^\infty P[n=k](1/2)\max _{i=1...k}(1+E[X_i])�$$ where $$\max _{i=1...k}(1+E[X_i])$$ is the maximum expectation for each child and factor of 1/2 appears as we assume 50-50 chance of being happy or sad. Are there such situations in literature?

Update: Since becoming familiar with multi type branching processes, I formatted my problem as follow:

I am considering a branching process with two types. I shall designate them with 1 and 2. The branching process follows Poisson distribution. So to that end I may write the probability generating functions as: $$f^1(\mathbf s)=\sum^{\infty}_{k_1=0}p_{k_1}s^{k_1}=e^{\lambda(s_1-1)}$$ $$f^2(\mathbf s)=\sum^{\infty}_{k_2=0}p_{k_2}s^{k_2}=e^{\lambda(s_2-1)}$$ I have borrowed this notation from the fact that, $$f^i({\mathbf s})=\sum_{\mathbf k}p_i(\mathbf k)s^{k_1}_{1}s^{k_2}_{2}\cdots s^{k_m}_{1}$$ Now I wonder how shall I proceed to calculate the extinction probability from here?

• I think you need two generating functions $H$ and $S$ in two variables: $h$ and $s$. Try googling multi-type Galton Watson process. – Anthony Quas Oct 5 '18 at 13:20
• This situation is definitely in the literature; these are typically called multi-type branching processes (rather than Galton-Watson processes). You can see an overview here, or check out the book by Mode on multitype processes for more information. – Marcus M Oct 6 '18 at 15:20
• Is it on purpose that sad faces only have sad children ? According to your description, it seems that it is not. If so, maybe you should arrange the picture. – M. Dus Oct 8 '18 at 17:16
• I don't follow your update; it appears from what you wrote that each type can only have children of its own type (i.e. $f^i$ only depends on $s_i$). Is this what you wanted? – Marcus M Oct 11 '18 at 21:47