  We consider random walks $\lambda$-biased towards the root on a Galton-Watson
tree, whose offspring distribution $(p_k)_{k\geq 1}$ is non-degenerate and has
finite mean $m>1$. In the transient regime $0<\lambda<m$, the loop-erased
trajectory of the biased random walk defines the $\lambda$-harmonic ray, whose
law is the $\lambda$-harmonic measure on the boundary of the Galton-Watson
tree. We answer a question of Lyons, Pemantle and Peres by showing that the
$\lambda$-harmonic measure has a.s. strictly larger Hausdorff dimension than
the visibility measure, which is the harmonic measure corresponding to the
simple forward random walk. We also prove that the average number of children
of the vertices along the $\lambda$-harmonic ray is a.s. bounded below by $m$
and bounded above by $m^{-1}\sum k^2 p_k$. Moreover, at least for $0<\lambda
\leq 1$, the average number of children of the vertices along the
$\lambda$-harmonic ray is a.s. strictly larger than that of the
$\lambda$-biased random walk trajectory. We observe that the latter is not
monotone in the bias parameter $\lambda$.
