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The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximumnumber of edges in any graph on $n$ vertices containing no $H$ as a subgraph.Let $P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the staron $\ell$ vertices and $k_1P_{\ell}\cup k_2S_{\ell-1}$ denote the path-starforest with disjoint union of $k_1$ copies of $P_{\ell}$ and $k_2$ copies of$S_{\ell-1}$. In 2013, Lidick\'y et al. first considered the Tur\'an number of$k_1P_4\cup k_2S_3$ for sufficiently large $n$. In 2022, Zhang and Wang raiseda conjecture about the Tur\'an number of $k_1P_{2\ell}\cup k_2S_{2\ell-1}$. Inthis paper, we determine the Tur\'an numbers of $P_{\ell}\cup kS_{\ell-1}$,$k_1P_{2\ell}\cup k_2S_{2\ell-1}$, $2P_5\cup kS_4$ for $n$ appropriately large,which implies the conjecture of Zhang and Wang. The corresponding extremalgraphs are also completely characterized.

Some results on the Tur\'an number of $k_1P_{\ell}\cup k_2S_{\ell-1}$