  Labeled infinite trees provide combinatorial interpretations for many integer
sequences generated by nested recurrence relations. Typically, such sequences
are monotone increasing. Several of these sequences also have straightforward
descriptions in terms of how often each value in the sequence occurs. In this
paper, we generalize the most classical examples to a larger family of
sequences parametrized by linear recurrence relations. Each of our sequences
can be constructed in three different ways: via a nested recurrence relation,
from labeled infinite trees, or by using Zeckendorf-like strings of digits to
describe its frequency sequence. We conclude the paper by discussing the
asymptotic behaviors of our sequences.
