Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002

 

 

Calculus on a tree with applications

 

Igor Urbiha

 

Department of  Mathematics, Faculty of Science and  Mathematics, University of Zagreb, Bijenička c. 30, HR-10002 Zagreb, Croatia

 

 

 

We describe simple algebraic operations with node-weighted complete infinite binary trees. A node-weighted binary tree consists of a binary tree (its frame) and a function assigning to each node of the tree a number (its entry). (Note the analogy with an array and its entries.) Some particular node-weighted complete infinite binary trees can be described implicitly by algebraic-like equations which also exploit structure of a tree. Sometimes some deeper structural properties can be read out directly from such equations. As an illustration of this
formalism we give some structural and numerical properties of the Stern-Brocot tree and their relations to hyperbinary partition function and some number theoretic functions studied by Eisenstein, Stern, de Rham, Carlitz, Dijkstra.

 

1 D. Svrtan, I. Urbiha, Calculus on a tree with applications to the Stern-Brocot tree., submitted.

2 D. H. Lehmer, On Stern's diatomic series, Amer. Math. Monthly 36(1) (1929), 59-67.

3 M. A. Stern, Ueber eine zahlentheoretische Funktion, Journal für die reine und angewandte Mathematik 55 (1858) 193-220.

4 R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994.