The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970:
I learned of it from Joe Roberts' book Elementary number theory: A problem oriented approach, published in 1977 by MIT Press. In 2005 it was published again by Robert C. Alperin:
He seems to say in a footnote on the first page that he didn't know that someone had discovered this before him until a referee pointed it out. That surprised me. Maybe because of Roberts' book, I had thought this was known to all who care about such things.
If we identify a Pythagorean triple with a rational point (or should I say a set of eight points?) on the circle of unit radius centered at $0$ in the complex plane, then we can say the set of all nodes in that tree is permuted by any linear fractional transformation that leaves the circle invariant. The function $f(z) = (az+b)/(bz+a)$, where $a$ and $b$ are real, leaves the circle invariant and fixes $1$ and $-1$. There are also LFTs that leave the circle invariant and don't fix those two points.
Are there any interesting relationships between the geometry of the LFT's action on the circle and that of the permutation of the nodes in the tree?
(Would it be too far-fetched if this reminds me of the recent discovery by Ruedi Suter of rotational symmetries in some well-behaved subsets of Young's lattice?)