> Equality and ordinality are not the same thing.  It should be
> perfectly reasonable to hand someone a set of unique objects sorted by
> some non-unique attribute of the elements of the set.

Sure, it's perfectly reasonable but it implies a different data
structure. It sounds like you want a data structure that is both a set
with respect to its default notion of equality (and hashing) and a
sorted sequence with respect to some custom ordering. That implies two
data structures, not one. If your tree is constructed based on the
custom ordering, then you cannot use the same tree for fast lookups
based on the default notion of equality. You can only do slow
linear-time lookups by traversing the tree's nodes exhaustively.

-Per

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