  We apply a tree-based methodology to solve new, very broadly defined families
of nested recursions of the general form R(n)=sum_{i=1}^k R(n-a_i-sum_{j=1}^p
R(n-b_{ij})), where a_i are integers, b_{ij} are natural numbers, and k,p are
natural numbers that we use to denote "arity" and "order," respectively, and
with some specified initial conditions. The key idea of the tree-based solution
method is to associate such recursions with infinite labelled trees in a
natural way so that the solution to the recursions solves a counting question
relating to the corresponding trees. We characterize certain recursion families
within R(n) by introducing "simultaneous parameters" that appear both within
the recursion itself and that also specify structural properties of the
corresponding tree. First, we extend and unify recently discovered results
concerning two families of arity k=2, order p=1 recursions. Next, we
investigate the solution of nested recursion families by taking linear
combinations of solution sequence frequencies for simpler nested recursions,
which correspond to superpositions of the associated trees; this leads us to
identify and solve two new recursion families for arity k=2 and general order
p. Finally, we extend these results to general arity k>2. We conclude with
several related open problems.
