I was reading Tavella Randall book and their explanation why binomial trees are a particular example of finite differences. I started having additional questions. So, they way they do that is saying that if there is a certain relation between $\Delta t$ and $\Delta x$, then the explicit finite difference scheme can be viewed as a binomial model. But how about stability? Is it always when I eliminate the second coefficient in my finite differences I will have a stable scheme? What is the stability then for the binomial trees, is there such a concept applied here as well?
Also, is there anything can be said about implicit method application here? Can binomial trees be stable and unstable?