  Burr, Erd\H{o}s, Faudree, Rousseau and Schelp initiated the study of Ramsey
numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all
odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices
and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum
positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$,
as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear.
In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and
$n \ge 50m$. Combining this with a result of Faudree, Lawrence, Parsons and
Schelp yields $n_0(m)$ is bounded between two linear functions, thus
identifying $n_0(m)$ up to a constant factor.
