We study a system of simple random walks on $Td,n=Vd,n,Ed,n)$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda$) particles at each site, independently, with one additional particle planted at some vertex $o$. Initially all particles are inactive, except for the ones which are placed at $o$. Active particles perform (independent) $ t \in \mathbb{N} \cup {\infty } $ steps of simple random walk on the tree. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $Rt$ be the set of vertices which are visited by the process. Let $\mathcal{S}(\mathcal{T}{d,n}) := \inf {t:\mathcal{R}t = \mathcal{V}{d,n} } $.Letthecovertime$\mathrm{CT}(\mathcal{T}{d,n})$ be the first time by which every vertex was visited at least once, when we take $t=\infty$. We show that there exist absolute constants, $c,C>0$ such that for all $d\ge 2$ and all $\lambda = \lambda_n $ which does not diverge nor vanish too rapidly, with high probability $c\le \lambda S(Td,n)/n\log (n/\lambda )\le C$ and $\mathrm{CT}(Td,n)\le 34\log |Vd,n|$.