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My Presentation on Row Sums of Calkin-Wilf Trees
1. ROW SUMS OF MUTATED CALKIN-
WILF TREES
Sophia Wang
2. Abstract
This paper investigated the relationship between sums of distinct Calkin-Wilf trees, in
the hopes of determining a formula to compute the sum of all the terms in any given row
of a tree with any roots. By exploring the fundamental Calkin-Wilf tree of root 1/1,
patterns were found for this specific set, and a simple equation for the sum was produced.
Sn=3⁄2 n^2 - ½ denotes the relations of the components of the tree, with Sn as the sum of
the terms in row n. This formula was used to expand on the research problem, as attempts
were tried-- employing different roots, calculating each row’s sum, and analyzing the data.
Ultimately, the results did not yield a possible formula for row sums, because such
mutated trees manipulated the patterns that were previously apparent in the C (1, 1) tree.
The research hypothesis was disproved with such investigation.
3. Introduction
The Calkin-Wilf forest is a set in which its individual trees have unique patterns that
depend entirely on the initial parent rational root of u/v.
● The starting point is a chosen number, and as the rows continue, the number of
branches grows exponentially. Each new term in the tree has its own two branches,
each of whom has its own as well.
○ Let the term right child define the value of the branch on the right side of the initial.
○ Let the term left child define the value of the branch on the left side of the initial root.
● Each branch stems out into an infinite series of double branches
○ each of which has a left child of u/(u+v) and right child of (u+v)/v.
● This was originally established as a method of depicting the entirety of rational roots.
○ Academic significance in this field of number theory and would allow other
scientists to solve further problems.
6. Significance
The Calkin-Wilf tree is fundamental in its portrayal of the infinite set of rational roots.
As the branches span on, there is really no end, which makes the question of its sums even
more significant.
What happens to each row’s sum as it continues to grow?
Would it be possible that every row would result in a computable sum?
Or would it eventually spiral off into infinity like the branches themselves?
These questions require investigation, which is the purpose of this research paper.
With the found formula, other research can be conducted to expand on such principles.
7. Methodology
● Let the notation, (u,v) represent a Calkin-Wilf tree with root of u/v.
● The basic Calkin-Wilf tree with root of 1/1 will span infinitely to present all possible
rational roots,
○ its specific branches are left child of x/(x+1) and
○ right child of x+1.
The investigated problem focuses on the row sums of a (u, v) tree, specifically the method
of calculating such sums.
● The methods used in this experiment consist of proving the row sums of the calkin-
wilf tree with roots (1, 1), and computing the row sums of mutated forms.
8. Methodology Continued
The fundamental Calkin-Wilf tree with root of 1/1 branches out to:
● left child of 1/(1+1) and
● right child of (1+1)/1.
Those values are contingent to the first row only, but a generalized version would be
left child 1/(x+1) and right child x+1.
We start with the following simple results pertaining specifically to C(1, 1). This basic
case is the first step in computing the row sums of a general mutated tree.
9. Lemma 4.1. If a number x is in row n then 1/x is also in row n.
Proof. C(1, 1) has a special relationship in which there is type of symmetry, marked by the
two branches that span out in Row 1. This is due to the equivalence of u and v.
The left child, originally u/u+v is now u/(u+u).
Similarly, the right child, originally (u+v)/u is now (u+u)/u.
→ The left child and right child are now reciprocals of each other.
This is applicable across the branches, with the pattern of the left child of the previous left
child equalling the reciprocal of the right child of the previous right child that came from
the same previous child.
→ In other words, the structure is split in the middle, and the terms matching up from the
center are reciprocals of each other.
12. General (u, v) case
The first 4 rows of generalized form of Calkin-Wilf tree, with starting root of
arbitrary fraction uv, or (u, v)
13. Observations
One notable attribute is how this fits into the original tree. Recall that one characteristic of
the C(1, 1) Calkin-Wilf tree is the inclusion of all positive, reduced fractions. Given this
property, it is certain that u/v is one of the terms of a branch in the infinite series. This
allows the reasonable assumption that the row sums of the (u, v) tree must be equal to or
less than the row sums of the C(1, 1) tree.
Furthermore, another big difference is the lack of reciprocal pairs in each row, since
essentially the tree is cut and separated from its “sibling” branch-- that is, the selected
point on the tree would be the new starting point, and thus its previous origins are lost, and
as too is the branch that originated from the same previous fraction.
14. Row Sums
As the rows span on, the fractions become more convoluted, with no apparent
relationship linking the transitions. The row sums did not divide well between each
other, nor were the results ideal when they were multiplied with, or subtracted from,
each of the other rows.
15. Case example: C(1, 2)
● Not friendly terms
● Regression line found through plotting
● However, when further cases were studied, and u and v
values were changed, the regression line changed as well
○ Close approximations, but nothing precise could be
determined
16. Discussion
● The lack of the symmetry in C(u,v) that was present in C(1,1) was a contribution to
the abstrusity of the sums.
● Each row sum is tedious to compute, and yields no relation to the rest.
○ Thus, there exists no apparent formula to compute the sum of any row in a
Calkin-Wilf tree of arbitrary u and v values-- for now.
● It is unfortunate that this research was not able to produce a conclusive formula, but it
was able to disprove the original hypothesis.
17. Discussion Continued
● This paper explored the notions of the Calkin-Wilf tree extremely thoroughly
○ showcases its merits, and unfortunate flaws.
○ establishes the significance of C (1, 1), whose symmetry allowed a pattern to
form. For that Calkin-Wilf tree of root 1/1, a formula was devised and proved.
○ The relationships between the branches in each row were more apparent in such
a basic case.
○ Tor more complex starting terms, the pattern is indiscernible due to its abrupt
beginnings.
18. Further Investigations
It is noticeable that in the regression for C (1,2), the row sums are roughly doubling.
○ If the nth row sum in C(1,2) were divided by 2^n it is possible to find it
approaching a limit
● It is definitely a valuable source of further investigation, though the results may be
very obscure and may not contribute much to the understanding of this notion.
■ Perhaps a computer program may aid this exploration, allowing the large
rational numbers to be more easily manipulated
19. References
[1] M. Aigner and G. Ziegler, Proofs from The Book, fourth ed., Springer-Verlag, Berlin, 2010.
[2] B. Bates, M. Bunder, and K. Tognetti, Linking the Calkin-Wilf and Stern-Brocot trees,
European J. Combin. 31 (2010) no. 7, 1637-1661
[3] B. Bates and T. Mansour,The q-Calkin-Wilf tree, J. Combin. Theory Ser. A 118(2011), no. 3
1143-1151
[4] S. P. Glasby,Enumerating the rationals from left to right, Amer. Math. Monthly118(2011) no.
9, 830-835.
[5] Sandie Han, Ariane M. Masuda, Satyanand Singh, Johann Thiel. The (u, v)-Calkin-Wilf
Forest. 6 Nov 2014.
[6] Melvyn B. Nathanson, Free monoids and forests of rational numbers. 9 Jun 2014. Retrieved
online. http://arxiv.org/abs/1406.2054v2
[7] Calkin, Neil. Recounting the Radicals. 6 July 1999. Retrieved online.
Special thanks to Dr. Shapovalov!!!!
