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\begin{abstract}
This paper is concerned with a family of $k$-ary meta-Fibonacci sequences
  described by the recurrence relation
\[
a(n) = \sum_{i=1}^k a(n{-}i - (s{-}1) - a(n{-}i)),
\]
where $s$ may be any integer, positive or negative.
If $s \ge 0$, then the initial conditions are
  $a(n) = 1$ for $1 \le n \le s+1$ and
  $a(n) = n-s$ for $s+1 < n \le s+k$.
If $s \le 0$, then the initial conditions are
  $a(n) = n$ for $1 \le n \le k(-s+1)$.
We show that these sequences arise as the solutions of two natural counting
  problems: The number of leaves at the largest level in certain
  infinite $k$-ary trees, and (for $s \ge 0$)
  certain restricted compositions of an integer.
For this family of generalized meta-Fibonacci sequences and two
  families of related sequences we derive combinatorial bijections,
  ordinary generating functions,
  recurrence relations, and asymptotics ($a(n) \sim n(k-1)/k$).
We also show that these sequences are related to a
  ``self-describing" sequence of Cloitre and Sloane.
\end{abstract}

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