We extend some results of Itai Benjamini and Yuri Lima (see [1]). In this paper they consider a binary tree Tn 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 pi 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 Tn as n goes to infinity. Here we want to study the more general case of a Galton-Watson tree and a z-ary tree.