Stern–Brocot Tree Explorer

The Stern–Brocot tree is the binary tree of every positive rational in lowest terms, each appearing exactly once. Start with the boundary ⁰⁄₁ and ¹⁄₀. At each step the tree inserts the mediant ^a + c⁄_b + d between the two current neighbours — and a mediant is automatically the fraction with the smallest denominator that lies strictly between them. Type any positive real and this tool walks the tree, giving you the L/R path, every convergent, and the sharpest rational approximation for a chosen denominator ceiling.

Target value Max denominator Max depth

Best approximation

Descent path (L = go left, R = go right)

Convergents from the continued fraction

k	a_k	p_k/q_k	value	error	\|x − p/q\|·q²

The |x − p/q| · q² column is Hurwitz’ test: an infinite subsequence of convergents keeps it below 1 / √5 ≈ 0.447 for irrational x.

Path down the tree (mediant sequence)

Why the mediant is optimal

A fraction ^p⁄_q is a best rational approximation of the second kind to x if no other ^p′⁄_q′ with q′ ≤ q satisfies |q′x − p′| < |qx − p|. Every best approximation of the second kind (with a mild exception at the head) is a convergent of x’s continued-fraction expansion — and every convergent is a node of the Stern–Brocot tree encountered along the descent toward x. That’s why the R/L path you see below is really the continued fraction in disguise: each run of L’s (or R’s) of length a_k corresponds to the partial quotient a_k.