  Chordal graphs are characterized as the intersection graphs of subtrees in a
tree and such a representation is known as the tree model. Restricting the
characterization results in well-known subclasses of chordal graphs such as
interval graphs or split graphs. A typical example that behaves computationally
different in subclasses of chordal graph is the \textsc{Subset Feedback Vertex
Set} (SFVS) problem: given a graph $G=(V,E)$ and a set $S\subseteq V$, SFVS
asks for a minimum set of vertices 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 $n^{O(\ell)}$. 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 directed path graphs that form a proper subclass of undirected path
graphs and graphs of mim-width one.
