  An infection spreads in a binary tree of height n as follows: initially, each
leaf is either infected by one of k states or it is not infected at all. The
infection state of each leaf is independently distributed according to a
probability vector p=(p_1,...,p_{k+1}). The remaining nodes become infected or
not via annihilation and coalescence: nodes whose two children have the same
state (infected or not) are infected (or not) by this state; nodes whose two
children have different states are not infected; nodes whose only one of the
children is infected are infected by this state. In this note we characterize,
for every p, the limiting distribution at the root node of the tree as the
height n goes to infinity.
  We also consider a variant of the model when k=2 and a mutation can happen,
with a fixed probability q, at each infection step. We characterize, in terms
of p and q, the limiting distribution at the root node of the tree as the
height n goes to infinity.
  The distribution at the root node is driven by a dynamical system, and the
proofs rely on the analysis of this dynamics.
