Unrooted binary tree enumeration problem on unrooted binary trees is to count the number of trees with "n" labeled leaves and unlabeled internal nodes. An unrooted binary tree on "n" labeled leaves can be formed by connecting the "n"th leaf to a new node in the middle of any of the edges of an unrooted binary tree on "n" − 1 labeled leaves. There are 2"n" − 5 edges at which the "n"th node can be attached; therefore, the number of trees on "n" leaves is larger than the number of trees on "n" − 1 leaves by a factor of 2"n" −
Combinatorial number system "C"<nowiki>'</nowiki>, since they have the same largest part 9, but the next largest part 6 of "C" is less than the next largest part 7 of "C"<nowiki>'</nowiki>; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0). Another way to describe this ordering is view combinations as describing the "k" raised bits in the binary representation of a number, so that "C" = {"c"...,"c"} describes the number (this associates distinct numbers to "all" finite sets of natural numbers); then comparison of "k"-combinations can be done by comparing the associated binary numbers. In the example "C" and "C"<nowiki>'</nowiki> correspond to numbers 1001011001 =