  This paper is a study of weighted counting of the solutions of acyclic
conjunctive queries ($\ACQ$). The unweighted quantifier free version of this
problem is known to be tractable (for combined complexity), but it is also
known that introducing even a single quantified variable makes it $\sP$-hard.
We first show that weighted counting for quantifier-free $\ACQ$ is still
tractable and that even minimalistic extensions of the problem lead to hard
cases. We then introduce a new parameter for quantified queries that permits to
isolate large island of tractability. We show that, up to a standard assumption
from parameterized complexity, this parameter fully characterizes tractable
subclasses for counting weighted solutions of $\ACQ$ queries. Thus we
completely determine the tractability frontier for weighted counting for
$\ACQ$.
