Let
${{G}_{1}},\,{{G}_{2}},\,.\,.\,.\,,\,{{G}_{t}}$
be arbitrary graphs. The Ramsey number
$R\left( {{G}_{1}},\,{{G}_{2}},\,.\,.\,.,{{G}_{t}} \right)$
is the smallest positive integer
$n$
such that if the edges of the complete graph
${{K}_{n}}$
are partitioned into
$t$
disjoint color classes giving
$t$
graphs
${{H}_{1}},\,{{H}_{2}},\,.\,.\,.\,,\,{{H}_{t}}$
, then at least one
${{H}_{i}}$
has a subgraph isomorphic to
${{G}_{i}}$
. In this paper, we provide the exact value of the
$R({{T}_{n}},\,{{W}_{m}})$
for odd
$m,\,n\,\ge \,m-1$
, where
${{T}_{n}}$
is either a caterpillar, a tree with diameter at most four, or a tree with a vertex adjacent to at least
$\left\lceil \frac{n}{2} \right\rceil \,-\,2$
leaves, and
${{W}_{n}}$
is the wheel on
$n\,+\,1$
vertices. Also, we determine
$R\left( {{C}_{n}},\,{{W}_{m}} \right)$
for even integers
$n$
and
$m,\,n\,\ge \,m\,+\,500$
, which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper.