I have opened a question here, but in this post I am going to expand it as I think the problem is more conceptual and relevant to this page.

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?