  For the critical Galton--Watson process with geometric offspring
distributions we provide sharp barrier estimates for barriers which are (small)
perturbations of linear barriers. These are useful in analyzing the cover time
of finite graphs in the critical regime by random walk, and the Brownian cover
times of compact two dimensional manifolds. As an application of the barrier
estimates, we prove that if $C_L$ denotes the cover time of the binary tree of
depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log
L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (1991),
Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent
article we use these barrier estimates to prove tightness of the Brownian cover
time for the two-dimensional sphere.
