We study the asymptotic behaviour of once-reinforced biased random walk
(ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk
has a bias towards or away from the root. We prove that in the setting of
multiplicative once-reinforcement the ORbRW can be recurrent even when the
underlying biased random walk is ballistic. We also prove that, on
Galton-Watson trees without leaves, the speed is positive in the transient
regime. Finally, we prove that, on regular trees, the speed of the ORbRW is
monotone decreasing in the reinforcement parameter when the underlying random
walk has high speed, and the reinforcement parameter is small
