To add to what others are saying about cyanobacteria and similar being the primary source of oxygen, you also have to consider that there is a HUGE amount of oxygen in the atmosphere. Even if oxygen production were ceased suddenly, it would take ages to get through it all.
This article argues that, at current world population, it would take a full 4000 years to consume it all:
www.scienceinschool.org/content/world-without-trees
As others say, there are assumptions built into your question that are false – the oxygen level does change (albeit not drastically), and ‘most’ trees don’t shed their leaves in the winter. The two hemispheres have opposite seasons so for every* (not 1:1 but close enough for comparison) tree that loses its leaves, another in the opposite side takes its place. In addition, most of the oxygen in the world comes from the oceanic plankton – after all, it makes up most of the surface of the planet.
It does. There is a ‘peak’ around August and a ‘valley’ around March. The thing is there is A LOT of oxygen in the atmosphere, and plenty of things that make it besides trees, so the variation is like 0.01-0.02% of the total atmosphere. In other words it isn’t something you notice outside of some pretty precise measurements.
The total mass of Earth’s atmosphere – one fifth of which is oxygen – is approximately 5×10^18 kg, or roughly a thousand times the Earth’s total living biomass (most of which *isn’t* trees).
Also, a bunch of those trees are in the southern hemisphere, where it’s currently summer.
Long story short, there’s a lot more air than there’s trees.
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