  We prove that the bisimulation-invariant fragment of weak monadic
second-order logic (WMSO) is equivalent to the fragment of the modal
$\mu$-calculus where the application of the least fixpoint operator $\mu
p.\varphi$ is restricted to formulas $\varphi$ that are continuous in $p$. Our
proof is automata-theoretic in nature; in particular, we introduce a class of
automata characterizing the expressive power of WMSO over tree models of
arbitrary branching degree. The transition map of these automata is defined in
terms of a logic $\mathrm{FOE}_1^\infty$ that is the extension of first-order
logic with a generalized quantifier $\exists^\infty$, where $\exists^\infty x.
\phi$ means that there are infinitely many objects satisfying $\phi$. An
important part of our work consists of a model-theoretic analysis of
$\mathrm{FOE}_1^\infty$.
