How many plants are needed to keep someone alive?

[Brother Oni]

Well, surely you can just take the averages over a population (on both the plant and human side) and use those? @Grytorm: I'll have a closer look at the link you provided, but it's quite dense and I don't know when I'll have the time to do it. Thanks!

Page 5, table 2 states that the standard crewman needs 0.84kg of oxygen daily
Page 11, Table 6 states that the standard crewman generates 1.00kg of carbon dioxide daily

Note that this CO2 is generated purely from ingested carbohydrates - all the respired O2 is excreted as water (see the Krebs Cycle and oxidative phosphorylation for further details).

Making the assumption that 80% of a fresh plant is water, that leaves 20% of the plant made of other elements and compounds. Assuming that 40% of this remainder is carbon, that makes 8g of carbon per 100g of plant material. Given photosynthesis generates 6 O2 per 6 CO2, that's 8g of O2 per 100g of plant material.

About 50% of the generated O2 is consumed by the plant itself (link), so that's 4g O2 / 100g plant material grown available for people.

Dividing the daily needs by that generation rate:

(840g/4g)*100g = 21 kg of plant mass grown daily to sustain a single human.

The fastest growing plant I know is bamboo; from this paper, that type of bamboo accumulates ~80% of maximal biomass in the first 40 days of growth.
Reading the paper, a large fully grown tree of ~13m will have ~20 kg of above ground biomass in 40 days. Since biomass is typically measured dry, we don't have to factor in the whole plant mass:

20 kg * 40% carbon = 8kg of captured carbon in 40 days
8kg captured carbon = 8 kg of released oxygen per 40 days
8kg oxygen * 50% for plant use = 4kg per 40 days available for human use.
4kg / 40 days = 100g oxygen average per day available for human use per plant.

0.84 kg oxygen daily need / 0.1 kg average daily oxygen generation = 8.4 plants per person, rounding up to 9 plants.

Optimal distance for mature bamboo is ~1 m so each plant needs about ~1.22 m² of space (assuming you have to keep it away from the walls of the arboretum), making it 11m². Adding in the 13m height of the plant, that's 143m³ per person.

Note that growing bamboo has a number of specific requirements (they like lots of water and warmth), plus factoring in soil depletion, slow initial growing, re-preparing the land again for new growth and infrastructure to supply everything, you're going to need a lot more space than that. There's also the issue of dealing with 180kg of dry bamboo every 40 days.

Edible plants would have a slower rate of growth, but the biomass could be recycled more efficiently. This is the harvest index mentioned in Page 11 of the NASA document and goes into some depth of the various systems and other variables you'd have to take into account.

Aquatic plants and things like phytoplankton could be more efficient but have their own issues like getting the CO2 and O2 in and out of solution and circulation.
Note that there's a number of critical assumption (primarily that 40% of captured CO2 from photosynthesis ends up as structural material), so tweak numbers accordingly if you don't believe my assumptions.