  Consider biased random walks on two Galton-Watson trees without leaves having
progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and
$P_2$ are supported on positive integers and $P_1$ dominates $P_2$
stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than
the same on GW$(P_2)$ when the bias is larger than a threshold depending on
$P_1$ and $P_2$. This partially answers a question raised by Ben Arous,
Fribergh and Sidoravicius.
