  We consider the Williams Bjerknes model, also known as the biased voter model
on the $d$-regular tree $\bbT^d$, where $d \geq 3$. Starting from an initial
configuration of "healthy" and "infected" vertices, infected vertices infect
their neighbors at Poisson rate $\lambda \geq 1$, while healthy vertices heal
their neighbors at Poisson rate 1. All vertices act independently. It is well
known that starting from a configuration with a positive but finite number of
infected vertices, infected vertices will continue to exist at all time with
positive probability iff $\lambda > 1$. We show that there exists a threshold
$\lambda_c \in (1, \infty)$ such that if $\lambda > \lambda_c$ then in the
above setting with positive probability all vertices will become eventually
infected forever, while if $\lambda < \lambda_c$, all vertices will become
eventually healthy with probability 1. In particular, this yields a complete
convergence theorem for the model and its dual, a certain branching coalescing
random walk on $\bbT^d$ -- above $\lambda_c$. We also treat the case of initial
configurations chosen according to a distribution which is invariant or ergodic
with respect to the group of automorphisms of $\bbT^d$.
