  We investigate the rank of the average mixing matrix of trees, with all
eigenvalues distinct. The rank of the average mixing matrix of a tree on $n$
vertices with $n$ distinct eigenvalues is upper-bounded by $\frac{n}{2}$.
Computations on trees up to $20$ vertices suggest that the rank attains this
upper bound most of the times. We give an infinite family of trees whose
average mixing matrices have ranks which are bounded away from this upper
bound. We also give a lower bound on the rank of the average mixing matrix of a
tree.
