Ramsey numbers for trees

For $n\ge 5$ let $T_n^{\mathrm{\prime }}$ denote the unique tree on $n$ vertices with $\mathrm{\Delta }(T_n^{\mathrm{\prime }})=n-2$, and let $T_n^{\ast }=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\dots ,$ $v_{n-1}\}$ and $E=\{v_0{v}_1,\dots ,v_0{v}_{n-3},$ $v_{n-3}{v}_{n-2},v_{n-2}{v}_{n-1}\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n^{\mathrm{\prime }})$ and $r(G_m,T_n^{\ast })$, where $G_m$ is a connected graph of order $m$. As examples, for $n\ge 8$ we have $r(T_n^{\mathrm{\prime }},T_n^{\ast })=r(T_n^{\ast },T_n^{\ast })=2n-5$, for $n>m\ge 7$ we have $r(K_{1,m-1},T_n^{\ast })=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$or$m-1\nmid (n-3)$,for$m\ge 7$and$n\ge (m-3)^2+2$wehave$r(T_m^,T_n^)=m+n-3$or$m+n-4$accordingas$m-1\mid (n-3)$or$m-1\nmid (n-3)$.