  The structure of Markov equivalence classes (MECs) of causal DAGs has been
studied extensively. A natural question in this regard is to algorithmically
find the number of MECs with a given skeleton. Until recently, the known
results for this problem were in the setting of very special graphs (such as
paths, cycles, and star graphs). More recently, a fixed-parameter tractable
(FPT) algorithm was given for this problem which, given an input graph $G$,
counts the number of MECs with the skeleton $G$ in $O(n(2^{O(d^4k^4)} + n^2))$
time, where $n$, $d$, and $k$, respectively, are the numbers of nodes, the
degree, and the treewidth of $G$.
  We give a faster FPT algorithm that solves the problem in $O(n(2^{O(d^2k^2)}
+ n^2))$ time when the input graph is chordal. Additionally, we show that the
runtime can be further improved to polynomial time when the input graph $G$ is
a tree.
