Page 34 - MATINF Nr. 8

34 5 CONCLUSIONS

1. How many leaves of value F 1 are in a Fibonacci tree with root F n ? Solution. One

, which is equal to F n , that can be easily calculated. Note: Analogously for leaves
determine T F 1
of value F 0 .

2. A Fibonacci tree has x (where x is a number from the Fibonacci sequence) leaves of value
F 1 . Determine the value of the biggest number from the level y (where y value is less than the
= F n value is equal to the value from the tree’s
tree’s height). Solution. We know that T F 1
root, so the searched value is F n−y . Note: Analogously for leaves of value F 0 .

3. We know that a Fibonacci subtree which is directly connected to the root of the Fibonacci
tree has x leaves of value F 0 and y leaves of value F 1 (where x and y are numbers from the
Fibonacci sequence). Calculate the number of the leaves with values F 1 and F 0 from the other
Fibonacci Subtree directly connected to the root. Solution. We will decrease or increase

respectively the order of the Fibonacci numbers beginning from the two values inserted.

5 Conclusions

The results can be used to study the properties of k-Fibonacci-type trees or to find a visual
representation of Fibonacci and Lucas numbers’ partitions or k-Fibonacci and k-Lucas numbers’
partitions, projects considered very useful by the author because of their huge potential in
Theoretical Computer Science (especially in the development of new algorithms, more efficient
than the Backtracking Algorithms).

It can be also given as an interesting model for high-school/college teachers which want to
demonstrate the beauty of the Graph Theory to their students.

References

[1] T. Caba, R.I. Mihai, Generating a congruence (in Romanian), Argument 19 (2017), 11-12.

[2] J. Gielis, Inventing the Circle. The Geometry of Nature, Geniaal, Antwerpen, 2003.

[3] R.I. Mihai, A geometric approach to Fibonacci and Lucas sequences, Parabola 56(1) (2020),
1-5.

[4] R.I. Mihai, A relationship between Lucas and Fibonacci sequences (in Romanian), Argument,
to appear.

[5] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications,
Dover Publ., Mineola, New York, 2008.

[6] https://oeis.org/

[7] https://xlinux.nist.gov/dads/HTML/fibonacciTree.html