Don't know what is meant by a desert island. I will take it to mean a place where there is no one living, but has some grasses, other plants, and perhaps some birds and small animals.
Would probably bring a he and she pig. Choice of animals would depend upon the flora and fauna available on the island. But pigs are fairly hearty and are often raised in areas where other animals cannot make it.
Also, could cure a piggy with dried salt from the ocean..
Likewise him and her chickens (for meat and eggs).
That and the seafood I could catch, snag, or trap, and any other tasty animals like turtles, or wild birds or animals I could hunt, would yield the protein.
Next I would think I would need some sort of grain, like wheat. It would provide flour for bread and could turn it into a beer and if I could figure out how to distill the stuff a whiskey (one must have priorities, and if I am going to be stuck on a desert island would like a libation. Sure I could, and would use whatever fruits I could forage for the boozerie. Just when is it happy hour on a desert island anyway?).
Dairy is a tough one to supply, but might include a brace of sheep or goats (himness and herness or course) for that purpose and maybe a bit of meat when one died (sorry but in olden times that is when the poor folk ate their animals). Would supply milk and could make cheese.
Now we are into the fruit and veggie category. Again don't know what is available on the island (such as coconuts, yeah, coconuts can be fermented) but would agree with Vera, tomatoes definitely. They are a fruit that is fast growing, has a lot of vitamin C, and I like them.
Potatoes, OK. They provide starch and are versitile. Would probably choose a sweet tater for the sugar content (and they can be fermented, right?).
Okra. Love it and find it a versatile veggie. And they have very pretty flowers and am thinking about gumbo here (chicken, seafood from the sea,and oh yeah that leads me to the next one.)
Rice. Gotta have rice.
Whatever is left is probably herbs.
That would do it unless I could smuggle some peanuts into the mix. Very tasty and need peanut butter (the fifth food group).
Think I could deal with that mix. And probably get fat. Urp.
But on a desert island who would care?
And there would always be happy hour.