For n≥5, let T′n denote the unique tree on n vertices with Δ(T′n)=n−2, and let T*n=(V,E) be the tree on n vertices with V ={v0,v1,…,vn−1} and E={v0v1,…,v0vn−3,vn−3vn−2,vn−2vn−1}. In this paper, we evaluate the Ramsey numbers r(Gm,T′n) and r(Gm,T*n) , where Gm is a connected graph of order m. As examples, for n≥8 we have r(T′n,T*n)=r(T*n,T*n)=2n−5 , for n>m≥7 we have r(K1,m−1,T*n)=m+n−3 or m+n−4 according to whether m−1∣n−3 or m−1∤n−3 , and for m≥7 and n≥(m−3)2 +2 we have r(T*m,T*n)=m+n−3 or m+n−4 according to whether m−1∣n−3 or m−1∤n−3 .