  We study the multiplication operators on the weighted Lipschitz space
$\mathcal{L}_{\textbf{w}}$ consisting of the complex-valued functions $f$ on
the set of vertices of an infinite tree $T$ rooted at $o$ such that
$\sup_{v\neq o}|v||f(v)-f(v^-)|<\infty$, where $|v|$ denotes the distance
between $o$ and $v$ and $v^-$ is the neighbor of $v$ closest to $o$. For the
multiplication operator, we characterize boundedness, compactness, provide
estimates on the operator norm and the essential norm, and determine the
spectrum. We prove that there are no isometric multiplication operators or
isometric zero divisors on $\mathcal{L}_{\textbf{w}}$.
