Levenshtein (1966) proposed a class of single insertion/deletion correcting codes, based on the number-theoretic construction due to Varshamov and Tenengolt's (1965). We present several interesting results on the binary structure of these codes, and their relation to constrained codes with nulls in the power spectral density function. One surprising result is that the higher order spectral null codes of Immink and Beenker (1987) are sub-codes of balanced Levenshtein codes. Other spectral null sub-codes with similar coding rates, may also be constructed. We furthermore present some coding schemes and spectral shaping markers which alleviate the fundamental restriction on Levenshtein's codes that the boundaries of each codeword should be known before insertion/deletion correction can be effected.