  We consider a continuous time random walk on the rooted binary tree of depth
$n$ with all transition rates equal to one and study its cover time, namely the
time until all vertices of the tree have been visited. We prove that,
normalized by $2^{n+1} n$ and then centered by $(\log 2) n - \log n$, the cover
time admits a weak limit as the depth of the tree tends to infinity. The
limiting distribution is identified as that of a Gumbel random variable with
rate one, shifted randomly by the logarithm of the sum of the limits of the
derivative martingales associated with two negatively correlated discrete
Gaussian free fields on the infinite version of the tree. The existence of the
limit and its overall form were conjectured in the literature. Our approach is
quite different from those taken in earlier works on this subject and relies in
great part on a comparison with the extremal landscape of the discrete Gaussian
free field on the tree.
