The real numbers in this tree do not correspond to nodes, they correspond to infinite chains starting at the root. There are uncountably many of those. To prove that, you can mimic Cantor's diagonal argument. If the number of chains were countable you could list them C1,C2,…C1,C2,...
For one irrational number there is indeed a deterministic generating procedure we can use to generate an associated countably infinite set of rational numbers. So what? Will the countably infinite set of rational numbers generated for the next irrational you pick re-use any of the rational ...
The axiom actually covers the situation with several (even countably infinite) events, every pair of which are mutually exclusive. As long as this occurs, theprobability of the unionof the events is the same as the sum of the probabilities: P(E1UE2U . . . UEn) =P(E1) +P(E2) + ....