  We look at a family of meta-Fibonacci sequences which arise in studying the
number of leaves at the largest level in certain infinite sequences of binary
trees, restricted compositions of an integer, and binary compact codes. For
this family of meta-Fibonacci sequences and two families of related sequences
we derive ordinary generating functions and recurrence relations. Included in
these families of sequences are several well-known sequences in the Online
Encyclopedia of Integer Sequences (OEIS).
