What is Calkin Wilf Sequence?
A Calkin-Wilf tree (or sequence) is a special binary tree which is obtained by starting with the fraction 1/1 and adding a/(a+b) and (a+b)/b iteratively below each fraction a/b. This tree generates every rational number. Writing out the terms in a sequence gives 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, …The sequence has the property that each denominator is the next numerator.
The image above is the Calkin-Wilf Tree where all the rational numbers are listed. The children of a node a/b is calculated as a/(a+b) and (a+b)/b.
The task is to find the nth rational number in breadth first traversal of this tree.
Input : 13 Output : [5, 3] Input : 5 Output : [3, 2]
Explanation: This tree is a Perfect Binary Search tree and we need floor(log(n)) steps to compute nth rational number. The concept is similar to searching in a binary search tree. Given n we keep dividing it by 2 until we get 0. We return fraction at each stage in the following manner:-
if n%2 == 0 update frac+=frac else update frac+=frac
Below is the program to find the nth number in Calkin Wilf sequence:
For n = 13,
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