Linking the Calkin-Wilf and Stern-Brocot trees
نویسندگان
چکیده
Links between the Calkin-Wilif tree and the Stern-Brocot tree are discussed answering the questions: What is the jth vertex in the nth level of the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the hyperbinary sequence thus answering a challenge raised in Quantum in September 1997. We also examine successors and predecessors in both trees.
منابع مشابه
Christoffel Words and the Calkin-Wilf Tree
In this note we present some results on the Calkin-Wilf tree of irreducible fractions, giving an insight on the duality relating it to the Stern-Brocot tree, and proving noncommutative versions of known results relating labels of the CalkinWilf trees to hyperbinary expansions of positive integers. The main tool is the Christoffel tree introduced in a paper by Berstel and de Luca.
متن کاملAn arborist’s guide to the rationals
There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as ‘transpose shadows’ of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.
متن کاملRecounting the Rationals: Twice!
We derive an algorithm that enables the rationals to be efficiently enumerated in two different ways. One way is known and is credited to Moshe Newman; it corresponds to a deforestation of the so-called Calkin-Wilf tree of rationals. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In ...
متن کاملSturmian words and the Stern sequence
Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of characteristic Sturmian words. A natural arithmetization of the theory is obtained by representing central and Christoffel words by irreducible fractions labeling respectively two binary trees, the Raney (or Calkin-Wilf) tree and the Stern-Brocot tree. ...
متن کاملBinary partitions of integers and Stern-Brocot-like trees
We study partitions of an integer n in which all parts are powers of 2, no power of 2 appearing more than twice. If b(n) is the number of these, and B(n) = P j n b(j), then we nd (a) that gcd (b(n); b(n+ 1)) = 1, and that every coprime (i; j) occurs exactly once as a pair of consecutive values (b) the asymptotic behavior of B(t2), for xed 1 < t < 2, as n!1 (c) descriptions of b(n) in terms of t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Eur. J. Comb.
دوره 31 شماره
صفحات -
تاریخ انتشار 2010