The Stern-Brocot Tree Gary D. Knott Civilized Software Inc. 12109 Heritage Park Circle Silver Spring MD 20906 email:knott@civilized.com URL:www.civilized.com January 5, 2019 Excerpt from book Exercise 0.1: Let a, c ∈Z and b, d ∈Z + with a b < c d . Show that a b < a + c b + d < c d . The fraction a + c b + d is called the mediant of a b and c d . Solution 0.1: Given a b < c d with b, d ∈Z + , we have ad < bc. But then ad 2 < bcd, so bad + ad 2 < bad + bcd. And ad < bc implies bad < b 2 c, so bad + bcd < b 2 c + bcd. Thus bad + ad 2 < bad + bcd < b 2 c + bcd. And then, (b + d)ad < (a + c)bd < (b + d)bc, so ad < bd(a + c) (b + d) < bc, and hence ad bd < bd(a + c) bd(b + d) < bc bd , or equivalently, a b < a + c b + d < c d . [QED] We discover this proof by starting with a b < a + c b + d < c d and going “backwards” to arrive at ad < bc. (It is also true that a b = c d is equivalent to a b = a + c b + d = c d .) The Stern-Brocot tree is an explicit recursive mediant construction of the rational numbers. It was written about (as a sequence of sequences) by Moritz Stern in 1858 and in a book by Achille Brocot in 1861 concerning its application to gear train optimization in clock-building where best rational approximations are sought. The Stern-Brocot tree and other related trees expose inter- esting structural features of the rational numbers related to their continued-fraction expansions. We shall introduce the full Stern-Brocot tree T by first presenting the positive Stern-Brocot tree T + . The Stern-Brocot tree T + is a binary tree with the root 1 1 and the left anchor 0 1 and the right 1