So I've been reading about brunching processes and came across the following statement:
$Z_n$ is a Galton-Watson process. Let $x$=$E[Z_1]$ and $n>m$ then:
$E[Z_nZ_m] = \sum_{r} P(Z_m=r)$$E[rz_n|Z_m=r]$=$\sum_{r} P(Z_m=r)$$r^2x^{n-m}$= $x^{n-m}$$E[Z_m^2]$
I just can't seem to grasp why this is true, specifically the second equality. Any help would be appreciated. And please tell me if this doesn't make sense out of context.