Solving Non-homogeneous Nested Recursions Using Trees

TL;DR

This paper extends the combinatorial tree-based approach to solve non-homogeneous nested recursions by introducing a tree-grafting method, enabling solutions for various initial conditions.

Contribution

It introduces a novel tree-grafting technique to handle non-homogeneous nested recursions, expanding the applicability of tree-based interpretations.

Findings

Tree-grafting method effectively solves non-homogeneous recursions.

The approach generalizes solutions to various initial conditions.

Provides a combinatorial interpretation for non-homogeneous cases.

Abstract

The solutions to certain nested recursions, such as Conolly's C(n) = C(n-C(n-1))+C(n-1-C(n-2)), with initial conditions C(1)=1, C(2)=2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a k-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions.