This is an interesting idea, but it has problems. If we write out the rational number as a binary fraction, we can see that you are basically encoding the path as sequences of binary zeros, and the 'dots' are the ones: node 2.3.2.1 would encode as:
0.01001011
and the path is: x2xx3x21
This implies the sum of the node numbers equals the number of binary digits required to express the numerator. Clearly you are going to run out if you use a normal 32 or 64 bit int. In this sense it has the same problems as chelko's proposal.

For run, I analysed our portfolio hierarchy. We would need 115 bits to represent the numerator for 2344 portfolios.