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Let $G$ and $G_1, G_2, \ldots , G_t$ be given graphs. By $G\rightarrow (G_1,G_2, \ldots , G_t)$ we mean if the edges of $G$ are arbitrarily colored by $t$colors, then for some $i$, $1\leq i\leq t$, the spanning subgraph of $G$ whoseedges are colored with the $i$-th color, contains a copy of $G_i$. The Ramseynumber $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ suchthat $K_n\rightarrow (G_1, G_2, \ldots , G_t)$ and the size Ramsey number$\hat{R}(G_1, G_2, \ldots , G_t)$ is defined as $\min\{|E(G)|:~G\rightarrow(G_1, G_2, \ldots , G_t)\}$. Also, for given graphs $G_1, G_2, \ldots , G_t$with $r=R(G_1, G_2, \ldots , G_t)$, the star-critical Ramsey number $R_*(G_1,G_2, \ldots , G_t)$ is defined as $\min\{\delta(G):~G\subseteq K_r,~G\rightarrow (G_1, G_2, \ldots , G_t)\}$. In this paper, the Ramsey number andalso the star-critical Ramsey number of a forest versus any number of completegraphs will be computed exactly in terms of the Ramsey number of completegraphs. As a result, the computed star-critical Ramsey number is used to give atight bound for the size Ramsey number of a forest versus a complete graph.

On the star-critical Ramsey number of a forest versus complete graphs