... self-describing sequences and the Catalan family tree Theorem The full names of the members of the Catalan family are precisely the selfdescribing sequences In other words, they are the fixed points of the ... distribution of labels in the Catalan family tree and the tree of the sequences with unit increase Theorem For r = 0, , n, the number of n-th generation vertices in the tree of sequences with unit ... preserves the length of the sequences and also their prefix order, i.e., a b implies αa αb, for all sequences a and b in A In the sequel, we often deliberately blur the distinction between the vertices...