Do you think Fruitarianism is a healthy lifestyle to follow?

The only diet more deficient would be the breatharian diet.

With the botanical definition of fruit many nuts could be included as well.

While some fruitarians are more restrictive than others, in the broadest sense a fruitarian diet includes any plant food that doesn't involve killing the plants (which tends to mean you eat the seed-bearing parts of plants rather than the whole plant). So for the least-restrictive fruitarians that would include beans/legumes, nuts, seeds (sesame, chia, flax, etc.), things we normally think of as fruits (berries, melons, citrus, apples, pears, stone fruit, etc.), culinary vegetables that are botanical fruits (tomatoes, peppers, avocados, cucumbers, eggplant/aubergines, summer and winter squash, etc.), and even grains (although the harvesting methods involved for grains probably make that ethically problematic for many if not most ethical fruitarians).

Things that vegans eat that would be off-limits on a fruitarian diet include stalks (like asparagus, celery, chard), pretty much all cruciferous vegetables (broccoli, cabbage, brussels sprouts, kale, etc.), pretty much all root vegetables (potatoes, carrots, onions, radishes, sweet potatoes, parsnips, turnips, rutabagas, etc.), and pretty much all leafy greens (I suppose some fruitarians might eat the kinds of greens where you harvest individual leaves but the plant continues to grow and put out new leaves).

So for a less restrictive fruitarian (one who includes legumes, nuts, seeds, and grains), I don't think protein and fat would really be more of an issue than for a vegan. Of course, they would have the same B12 concerns that vegans face. Perhaps there are micronutrients, particularly minerals, available in green vegetables that might be hard to come by on a fruitarian diet.

For the more restrictive fruitarians who exclude some or all of the legume, nut, seed, and grain categories, ... I dunno. Seems like a catch-22 if your respect for all life, even plant life, means that you're denying yourself nutrition that's essential to your own life.