A variation of the Stern-Brocot tree
نویسنده
چکیده
The Stern-Brocot tree (or rather half of it) can be defined as follows. Start with two fractions 0/1 and 1/1, forming an ordered set S0. (Throughout this paper, “fraction” means “fraction in lowest terms”.) At stage k, (k = 1, 2, . . .), form a new set Sk by inserting between each pair of adjacent fractions in Sk−1, say p/q and r/s, the fraction (p+r)/(q+s). Name the (ordered) set of fractions that are introduced at this stage Rk. Thus R1 = (1/2), R2 = (1/3, 2/3), R3 = (1/4, 2/5, 3/5, 3/4), R4 = (1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5) etc. Rk has 2 k−1 elements. It is well known (see e.g. [1]) that every proper fraction appears (exactly once) in some Rk, and that adjacent fractions p/q, r/s satisfy
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