Fix $p \in (0,1)$ and suppose we have a sequence of random variables defined as follows: let $X_0 = 1$, and given $X_n$, we have the binomial distribution
$$X_{n+1} \sim \text{Bin}(X_n,p) + X_n.$$
I.e., $X_n$ describes the size of the $n$th generation of a Galton-Watson tree with distribution $\xi$: $\mathbb P (\xi = 1) = 1- p$ and $\mathbb P (\xi = 2) = p$.
The Kesten–Stigum theorem gives, for $\mu = \mathbb E (\xi) = 1 + p$, that
$$\frac {X_n} {\mu^n}$$
converges almost-surely to a random limit $Y$.
My question is: can we easily show the distribution of $Y$ is non-trivial?
(I.e., it is not a Dirac mass.)
[And can we say anything further about its distribution?]