Diseases transmission in a z-ary tree

We extend some results of Itai Benjamini and Yuri Lima (see \href{http://arxiv.org/pdf/1305.2610.pdf}{\cite{Benjamini}}). In this paper they consider a binary tree $\mathbb T_n$ of height $n$, each leaf is either infected by one of $k$ diseases or not infected at all. In other words, $x$ at generation $n$ is infected by the $i$-th infection with probability $p_i$ and sane with $p_{k+1}$. Moreover the infections are independently distributed for each leaf. Infections spread along the tree based on specific rules. In their paper they study the limit distribution of the root of $\mathbb T_n$ as $n$ goes to infinity. Here we want to study the more general case of a Galton-Watson tree and a $z$-ary tree.