  The Minimum Branch Vertices Spanning Tree problem aims to find a spanning
tree $T$ in a given graph $G$ with the fewest branch vertices, defined as
vertices with a degree three or more in $T$. This problem, known to be NP-hard,
has attracted significant attention due to its importance in network design and
optimization. Extensive research has been conducted on the algorithmic and
combinatorial aspects of this problem, with recent studies delving into its
fixed-parameter tractability.
  In this paper, we focus primarily on the parameter modular-width. We
demonstrate that finding a spanning tree with the minimum number of branch
vertices is Fixed-Parameter Tractable (FPT) when considered with respect to
modular-width. Additionally, in cases where each vertex in the input graph has
an associated cost for serving as a branch vertex, we prove that the problem of
finding a spanning tree with the minimum branch cost (i.e., minimizing the sum
of the costs of branch vertices) is FPT with respect to neighborhood diversity.
