We show that for $\lambda\in[0,{m_1}/({1+\sqrt{1-{1}/{m_1}}})]$, the biased random walk’s speed on a Galton–Watson tree without leaves is strictly decreasing, where $m_1\geq 2$. Our result extends the monotonic interval of the speed on a Galton–Watson tree.

We establish an invariance principle and a large deviation principle for a biased random walk ${\text{RW}}_\lambda$ with $\lambda\in [0,1)$ on $\mathbb{Z}^d$ . The scaling limit in the invariance principle is not a d-dimensional Brownian motion. For the large deviation principle, its rate function is different from that of a drifted random walk, as may be expected, though the reflected biased random walk evolves like the drifted random walk in the interior of the first quadrant and almost surely visits coordinate planes finitely many times.