Distinguishing partitions of complete multipartite graphs

A $\mathit{\text{distinguishing}}\mathit{\text{partition}}$ of a group $X$ with automorphism group $aut(X)$ is a partition of $X$ that is fixed by no nontrivial element of $aut(X)$. In the event that $X$ is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when $X$ is a complete multipartite graph with $m_1$ parts of size $n_1$ and $m_2$ parts of size $n_2$ for small $n_1$, $m_2$ and large $m_1$, $n_2$. A key tool in making the estimate is counting the number of trees of particular classes.