  Following the recent paper which initiated the study of colour isomorphism
problems for complete graphs, we obtain upper bounds for $f_2(n,H)$ for a
family of graphs $H$ obtained as the $K_0$-th rooted power of a balanced rooted
tree for some sufficiently large $K_0$. The proof uses the random polynomial
method of Bukh. We also obtain matching lower bounds for $1$-subdivisions of
the complete bipartite graph.
