  A Bayesian network is a directed acyclic graph that represents statistical
dependencies between variables of a joint probability distribution. A
fundamental task in data science is to learn a Bayesian network from observed
data. \textsc{Polytree Learning} is the problem of learning an optimal Bayesian
network that fulfills the additional property that its underlying undirected
graph is a forest. In this work, we revisit the complexity of \textsc{Polytree
Learning}. We show that \textsc{Polytree Learning} can be solved in $3^n \cdot
|I|^{\mathcal{O}(1)}$ time where $n$ is the number of variables and $|I|$ is
the total instance size. Moreover, we consider the influence of the number of
variables $d$ that might receive a nonempty parent set in the final DAG on the
complexity of \textsc{Polytree Learning}. We show that \textsc{Polytree
Learning} has no $f(d)\cdot |I|^{\mathcal{O}(1)}$-time algorithm, unlike
Bayesian network learning which can be solved in $2^d \cdot
|I|^{\mathcal{O}(1)}$ time. We show that, in contrast, if $d$ and the maximum
parent set size are bounded, then we can obtain efficient algorithms.
