The nodes of the Calkin–Wilf tree, when read in level order so that the elements in each level are read from left to right, produce the linear sequence of all possible positive rational numbers. Almost as if by magic, this construction guarantees every positive integer fraction to appear exactly once in this sequence. Even more delightfully, this construction makes every rational number to make its appearance in its lowest reduced form!

The tree is rooted at the number 1 (1/1), and any rational number expressed in simplest terms as the fraction a/b has as its two children the numbers a/(a + b) and (a + b)/b.

Your function should return the n:th element of this sequence. Notice, that once you reach the position n//2 + 1, the queue already contains the result you need, so you can save a hefty chunk of time and space by not finding any new values. Besides, the linked Wikipedia page and other sources provide additional shortcuts to jump into the given position faster than sloughing your way the hard way there one element at a time.

Input: Integer (int).

Output: Two integers.

Examples:

assert calkin_wilf(1) == (1, 1)
assert calkin_wilf(2) == (1, 2)
assert calkin_wilf(3) == (2, 1)
assert calkin_wilf(10) == (3, 5)

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen