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What is the smallest set of different crops a (vegan) human population could grow to sustain itself?
We'll assume they have an ideal environment, where anything and everything can grow and the technology to grow aquatic plants. It can be anything from baobab fruits to micro-algaes.
My first list contains soy beans, spirulina, barley, cabbage, bananas, and a few others. It keeps getting longer but I'd like to keep it as short as possible.
I should note that having the minimum required is a bad idea. If one disease or disaster wipes out a particular plant then you are in serious trouble.
Really you want at least 2 plant species providing for each requirement, which will also help make the diet more varied and interesting.
You should not overlook the humble potatoes: https://en.wikipedia.org/wiki/Potato#Nutrition
Other good choices include:
Barley. From your list above this is an excellent choice.
Beans. Black, Red and Soy beans are all excellent sources of calcium and iron. They can also be dried for long term storage.
Berries. Pretty much all non-poisonous berries are packed full of goodness.
Cantaloupe. An excellent source of vitamins A and C and potassium. It's a slightly upgraded banana which would also be a good choice.
Kale. Particularly rich in vitamins, minerals and fiber. Other cabbage varieties are also strong contenders.
You should look at some of the research that's been done on subsistence diets and on space exploration. Space Exploration is considering genetically engineering the crops though.
The agriculture of the Iroquois and other northeastern (American) Indians was based on the "Three Sisters": corn (maize to British/Europeans), beans, and squash. Of course they had other plants, cultivated and wild-gathered, as well as meat from hunting. I don't know whether the three alone would provide a complete diet, but it's a starting place, and has the benefit of tradition.
And of course the Irish survived mainly on potatos for quite some time, but this (https://en.wikipedia.org/wiki/Great_Famine_(Ireland) ) is a good example of why you wouldn't want to be dependent on a small set of plants.
