Cover time for the frog model on trees

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\mu$ on the full $d$-ary tree of height $n$. If $\mu =\mathrm{\Omega }(d^2)$, all of the vertices are visited in time $\mathrm{\Theta }(n\log n)$ with high probability. Conversely, if $\mu =O(d)$ the cover time is $\exp (\mathrm{\Theta }(\sqrt{n}))$ with high probability.