[seqfan] 2-3 sequence puzzle

[Vladimir Reshetnikov]

Dear Seqfans,

Here is a sequence consisting of all non-empty finite strings of digits 2
and 3 without any duplicates. It begins as:

2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332, 2222, 3222, 233, 333,
2322, 3322, 2223, 3223, 2232, 3232, 2323, 3323, 2332, 3332, 22222, 32222,
23222, 33222, 2233, 3233, 2333, 3333, 22322, 32322, 23322, 33322, 22223,
32223, 23223, 33223, 22232, 32232, 23232, 33232, 22323, 32323, 23323,
33323, 22332, 32332, 23332, 33332, 222222, 322222, 232222, 332222, 223222,
323222, 233222, 333222, 22233, 32233, 23233, 33233, 22333, 32333, 23333,
33333, 222322, ...

Can you guess the order in what they appear?

--
Vladimir