  The uniform spanning forest measure ($\mathsf{USF}$) on a locally finite,
infinite connected graph $G$ with conductance $c$ is defined as a weak limit of
uniform spanning tree measure on finite subgraphs. Depending on the underlying
graph and conductances, the corresponding $\mathsf{USF}$ is not necessarily
concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that
on $\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\mathsf{USF}$ is
concentrated on spanning trees if and only if $d \leq 4$. In this work we study
the $\mathsf{USF}$ associated with conductances induced by $\lambda$--biased
random walk on $\mathbb{Z}^d$, $d \geq 2$, $0 < \lambda < 1$, i.e. conductances
are set to be $c(e) = \lambda^{-|e|}$, where $|e|$ is the graph distance of the
edge $e$ from the origin. Our main result states that in this case
$\mathsf{USF}$ consists of finitely many trees if and only if $d = 2$ or $3$.
More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d
= 2$ or $3$, and infinitely many trees if $d \geq 4$. Our method relies on the
analysis of the spectral radius and the speed of the $\lambda$--biased random
walk on $\mathbb{Z}^d$.
