Let $s$ and $t$ be integers satisfying $s \geq 2$ and $t \geq 2$. Let $S$ be a tree of size $s$, and let $P_t$ be the path of length $t$. We show in this paper that, for every edge-colouring of the complete graph on $n$ vertices, where $n=224(s-1)^2t$, there is either a monochromatic copy of $S$ or a rainbow copy of $P_t$. So, in particular, the number of vertices needed grows only linearly in $t$.