> positions are encoded as strings over the alphabet on 2 bits

This is the most pedantic way of saying "binary 2-tuples" I've ever seen. Also for quadtrees this is inferior to base 4 because you can assume clockwise (or counter) ordering.

I don't think that's what they meant. It's the case you can use literal strings of bits to encode a (2^n)-tree node, so you use actual bitstring comparisons and operations to manipulate them. Rightshift gives you the parent and things like that.

I don't think this is something the article cares about, though.