The number 1 (expressed as a fraction 1/1) is placed at the root of the tree, and the location of any other number "a"/"b" can be found by computing gcd("a","b") using the original form of the Euclidean algorithm, in which each step replaces the larger of the two given numbers by its difference with the smaller number (not its remainder), stopping when two equal numbers are reached. A step of the Euclidean algorithm that replaces the first of the two numbers corresponds to a step in the tree from a node to its right child, and a step that replaces the second of the two numbers corresponds to a step in the tree from a node to its left child. The sequence of steps constructed in this way does not depend on whether "a"/"b" is given in lowest terms, and forms a path from the root to a node containing the number "a"/"b". This fact can be used to prove that each positive rational number appears exactly once in this tree.