the th value in the sequence is the value of the fusc function, named according to the obfuscating appearance of the sequence of values and defined by the recurrence relations with the base cases and . The th rational number in a breadth-first traversal of the Calkin–Wilf tree is the number . Thus, the diatomic sequence forms both the sequence of numerators and the sequence of denominators of the numbers in the Calkin–Wilf sequence. The function is the number of odd binomial coefficients of the form , , and also counts the number of ways of writing as a sum of powers of two in which each power occurs at most twice. This can be seen from the recurrence defining fusc: the expressions as a sum of powers of two for an even number either have no 1s in them (in which case they are formed by doubling each term in an expression for ) or two 1s (in which case the rest of the expression is formed by doubling each term in an expression for ), so the number of representations is the sum of the number of representations for and for , matching the recurrence. Similarly, each representation for an odd number is formed by doubling a representation for and adding 1, again matching the recurrence. For instance, 6 = 4 + 2 = 4 + 1 + 1 = 2 + 2 + 1 + 1 has three representations as a sum of powers of two with at most two copies of each power, so . Relation to Stern–Brocot tree The Calkin–Wilf tree resembles the Stern–Brocot tree in that both are binary trees with each positive rational number appearing exactly once. Additionally, the top levels of the two trees appear very similar, and in both trees, the same numbers appear at the same levels. One tree can be obtained from the other by performing a bit-reversal permutation on the numbers at each level of the trees. Alternatively, the number at a given node of the Calkin–Wilf tree can be converted into the number at the same position in the Stern–Brocot tree, and vice versa, by a process involving the reversal of the continued fraction representations of these numbers. However, in other ways, they have different properties: for instance, the Stern–Brocot tree is a binary search tree: the left-to-right traversal order of the tree is the same as