  A typical example that behaves computationally different in subclasses of
chordal graphs is the \textsc{Subset Feedback Vertex Set} (SFVS) problem: given
a vertex-weighted graph $G=(V,E)$ and a set $S\subseteq V$, the \textsc{Subset
Feedback Vertex Set} (SFVS) problem asks for a vertex set of minimum weight
that intersects all cycles containing a vertex of $S$. SFVS is known to be
polynomial-time solvable on interval graphs, whereas SFVS remains \NP-complete
on split graphs and, consequently, on chordal graphs. Towards a better
understanding of the complexity of SFVS on subclasses of chordal graphs, we
exploit structural properties of a tree model in order to cope with the
hardness of SFVS. Here we consider variants of the \emph{leafage} that measures
the minimum number of leaves in a tree model. We show that SFVS can be solved
in polynomial time for every chordal graph with bounded leafage. In particular,
given a chordal graph on $n$ vertices with leafage $\ell$, we provide an
algorithm for SFVS with running time $n^{O(\ell)}$. We complement our result by
showing that SFVS is \W[1]-hard parameterized by $\ell$. Pushing further our
positive result, it is natural to consider a slight generalization of leafage,
the \emph{vertex leafage}, which measures the smallest number among the maximum
number of leaves of all subtrees in a tree model. However, we show that it is
unlikely to obtain a similar result, as we prove that SFVS remains \NP-complete
on undirected path graphs, i.e., graphs having vertex leafage at most two.
Moreover, we strengthen previously-known polynomial-time algorithm for SFVS on
rooted path graphs that form a proper subclass of undirected path graphs and
graphs of mim-width one.
