  We study the computational complexity of counting constraint satisfaction
problems (#CSPs) whose constraints assign complex numbers to Boolean inputs
when the corresponding constraint hypergraphs are acyclic. These problems are
called acyclic #CSPs or succinctly, #ACSPs. We wish to determine the
computational complexity of all such #ACSPs when arbitrary unary constraints
are freely available. Depending on whether we further allow or disallow the
free use of the specific constraint XOR (binary disequality), we present two
complexity classifications of the #ACSPs according to the types of constraints
used for the problems. When XOR is freely available, we first obtain a complete
dichotomy classification. On the contrary, when XOR is not available for free,
we then obtain a trichotomy classification. To deal with an acyclic nature of
constraints in those classifications, we develop a new technical tool called
acyclic-T-constructibility or AT-constructibility, and we exploit it to analyze
a complexity upper bound of each #ACSPs.
