I'm not sure if they should or not, but I give both of them carrots every day for the antioxidants. Vlad gets 4 baby carrots a day and Barkly gets two.
Getting antioxidants every day is why I feed them veg puree, and let them have fruit at lunch. If it weren't for the antioxidants, I wouldn't feed them anything but meat. But after having so much cancer in our past dogs, I'm really afraid for them to not get antioxidants & green/yellow veg is the only way I know to do the best ones. I just prohibit things that are known as poisonous like onions, grapes, etc. and grains of any sort.