  A set of words, also called a language, is letter-balanced if the number of
occurrences of each letter only depends on the length of the word, up to a
constant. Similarly, a language is factor-balanced if the difference of the
number of occurrences of any given factor in words of the same length is
bounded. The most prominent example of a letter-balanced but not
factor-balanced language is given by the Thue-Morse sequence. We establish
connections between the two notions, in particular for languages given by
substitutions and, more generally, by sequences of substitutions. We show that
the two notions essentially coincide when the sequence of substitutions is
proper. For the example of Thue-Morse-Sturmian languages, we give a full
characterisation of factor-balancedness.
