One-One Mapping of Reals and Sequences

Is there a "canonical" one-to-one mapping between the reals on the
interval [0,1) and the set of semi-infinite binary sequences, taking
account of the fact that some reals have two representations?
W. Schneeberger mentioned in email that, since each real corresponding 
to a terminating binary expansion has two redundant non-terminating 
expansions, we can map these two representations down to the next 
level.  Thus, the sequences {x0111...} and {x1000...} are mapped to 
the distinct binary  reals {.x01000...} and {.x11000...} respectively.
The first few levels are
                                   1
                     01                          11
              001          011            101         111
          0001  0011    0101  0111    1001  1011   1101  1111

This is particularly convenient for binary sequences because each 
"level" of terminating representations has twice as many members 
as the preceding level, so it's easy to define a mapping that absorbs 
the double representations.  This seems like a good candidate for the 
canonical one-to-one mapping.

For other bases it's not quite as natural, but I suppose a similar 
method could be applied.  In base B, the first level has B-1 members, 
and each successive level has B times the previous level.  Therefore, 
it's necessary to carry down (2)(B-1) from the first to the second 
level, and (2+B)(B-1) from the second to the third level, and 
(2+B+B^2)(B-1) from the third to the fourth level, and so on.  At 
the (n+1)th level we need to carry (B^n + B - 2) sequences.  This 
leaves just B-1 unmapped reals (from the first level), which shouldn't 
be too difficult to absorb.