We present a sound and complete satisfiability-testing algorithm and its effective implementation for an alternation-free modal $\mu$-calculus with converse, where formulas are cycle-free, and which is interpreted over finite ordered trees. The time complexity of the satisfiability-testing algorithm is $2^{\mathcal{O}(n)}$ in terms of formula size $n$. The algorithm is implemented using symbolic techniques (BDD). We present crucial implementation techniques and heuristics that we used to make the algorithm as fast as possible in practice. Our implementation is available online, and can be used to solve logical formulas of significant size and practical value.