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Solving Non-Homogeneous Nested Recursions Using Trees.
- Source :
-
Annals of Combinatorics . Nov2013, Vol. 17 Issue 4, p695-710. 16p. - Publication Year :
- 2013
-
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, and its generalization to a similar k-term nested recursion, only apply to homogeneous recursions and only solve 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 also solves the given non-homogeneous recursion with various sets of initial conditions. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02180006
- Volume :
- 17
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Annals of Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 92014959
- Full Text :
- https://doi.org/10.1007/s00026-013-0202-9