A sharp estimate for cover times on binary trees

We compute the second order correction for the cover time of the binary tree of depth nn by (continuous-time) random walk, and show that with probability approaching 11 as nn increases, τcov=|E|[2log2⋅n−logn/2log2+O((loglogn)8)], thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree.