The reaping number ${𝔯}_{m,n}\left(𝔹\right)$ of a Boolean algebra $𝔹$ is defined as the minimum size of a subset $𝒜\subseteq 𝔹\setminus \left\{𝐎\right\}$ such that for each $m$-partition $𝒫$ of unity, some member of $𝒜$ meets less than $n$ elements of $𝒫$. We show that for each $𝔹$, ${𝔯}_{m,n}\left(𝔹\right)={𝔯}_{\lceil \frac{m}{n-1}\rceil ,2}\left(𝔹\right)$ as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every $k$-branching tree whose maximal nodes are coloured with $\ell$ colours contains an $m$-branching subtree using at most $n$ colours if and only if $\lceil \frac{\ell }{n}\rceil <\lceil \frac{k}{m-1}\rceil$.