Summary: ``Let $G$ be a graph of size ${n+1\choose2}$ for some integer $n\geq2$. Then $G$ is said to have an ascending star subgraph decomposition if $G$ can be decomposed into $n$ subgraphs $G_1,G_2,\cdots,G_n$ such that each $G_i$ is a star of size $i$ with $1\leq i\leq n$. We prove in this paper that a star forest with size ${n+1\choose2}$ possesses an ascending star subgraph decomposition under some conditions on the number of components or the size of components.''