  We develop a technique that provides a lower bound on the speed of transient
random walk in a random environment on regular trees. A refinement of this
technique yields upper bounds on the first regeneration level and regeneration
time. In particular, a lower and upper bound on the covariance in the annealed
invariance principle follows. We emphasize the fact that our methods are
general and also apply in the case of once-reinforced random walk. Durrett,
Kesten and Limic (2002) prove an upper bound of the form $b/(b+\delta)$ for the
speed on the $b$-ary tree, where $\delta$ is the reinforcement parameter. For
$\delta>1$ we provide a lower bound of the form $\gamma^2 b/(b+\delta)$, where
$\gamma$ is the survival probability of an associated branching process.
