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Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and avertex of degree at least three is a \emph{branch vertex} of $T$. A graph issaid to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an inducedsubgraph. In this paper, we study the spanning trees with a bounded number ofleaves and branch vertices of $K_ {1,4}$-free graphs. Applying the mainresults, we also give some improvements of previous results on the spanningtree with few branch vertices for the case of $K_{1,4}$-free graphs.

Spanning trees of $K_{1,4}$-free graphs with a bounded number of leaves and branch vertices