This article is from the Aquaria: Plants FAQ, by multiple authors.
No. The level of dissolved CO2 and oxygen are actually independent of
each other; high levels of both can exist at the same time.
Furthermore, if you have a set of healthy plants, they will be
saturating the water with oxygen on their own. The problem is that
many of the techniques used to increase oxygen content (airstones,
trickle filters, keeping the water moving at the surface) also cause
CO2 to diffuse out of the aquarium; i.e., if you turn off your
airstone in order to keep the CO2 in, you might also reduce your
oxygen content. The best solution is to keep the water moving at the
surface of the tank, but inject CO2 faster than it can escape, giving
you high levels of both CO2 and oxygen.
 
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