On a tree of rational functions related to continued fractions

In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example, the zero solutions of the denominators $q(x)$ are all real negative numbers and are dense in $(-\infty,-1]$. For $x>0$ functions are non intersecting and form a dense subset of $(0,1)$. Furthermore, when evaluating the tree for positive rational values, the tree contains every rational in $(0,1)$ exactly once if and only if $x\in \mathbb{N}$. For $x=1$, one finds back the classical Farey tree which is related to regular continued fractions. In the last part, we will make a similar tree in a similar way but for backward continued fractions. We highlight some similarities and differences.