Abstract: For a graph G, let $ \mu (G)=\min \{\max \{{d}(x),{d}(y)\}:x\ne y,xy\notin E(G), x,y\in V(G)\} $ if G is non-complete, otherwise, $ \mu (G)=+\infty. $ For a given positive number s, we call that a graph G satisfies Fan-type conditions if $ \mu (G)\geqslant s. $ Suppose $ \mu (G)\geqslant s, $ then a vertex v is called a small vertex if the degree of v in G is less than s. In this paper, we prove that for a forest F with m edges, if G is a graph of order $ n\geqslant |F| $ and $ \mu (G)\geqslant m $ with at most $ \max \{n-2m,0\} $ small vertices, then G contains a copy of F. We also give examples to illustrate both the bounds in our result are best possible.

Key words: Fan-type conditions, Forest, Copy, Small vertex