Some sufficient conditions for graphs to have component factors

Let $G$ denote a graph and $k\ge 2$ be an integer. A $\{K_{1,1},K_{1,2},\dots ,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of $\{K_{1,1},K_{1,2},\dots ,K_{1,k},\mathcal{T}(2k+1)\}$, where $\mathcal{T}(2k+1)$ is one special family of tree. In this paper, we put forward some sufficient conditions for the existence of $\{K_{1,1},K_{1,2},\dots ,K_{1,k},\mathcal{T}(2k+1)\}$-factors in graphs. Furthermore, we construct some extremal graphs to show that the main results in this paper are best possible.