Malnutrition can be a cause of tooth loss, among other things (such as osteoporosis, brittle nails, hair loss, etc.). However, just because one is anorexic doesn't mean that they don't binge/purge at all. I believe that there are two subsets of anorexia (restricting and purging) just like there are two subsets to bulimia (purging and non-purging).

More information about anorexia HERE.

More information about bulimia HERE.