In development...

Definition 1. The infinite planar binary tree is constructed by creating the root node, and adding two children to it, and inductively, adding two children to each node that has not yet children, ad infinitum. (Formulate this better)


Definition 2. The breadth-first, left-to-right enumeration of the nodes of this binary tree is accomplished by denoting the root node as 1, and then traversing each successive level (i.e. those nodes from which the distance to the root is equal) of the tree, from left to right, denoting each node with successive integers. 
                             1
             2                               3
     4               5               6               7
 8       9       10     11      12     13       14     15


Definition 3. The rotation of node n to right, indicated here with ρn (ρ = a small greek letter rho), is accomplished by severing the edge between its left child and the left child's right child, moving the left child (and the whole subtree under its left child) to the former position of the node n, moving the node n to the former position of its right child, which in turn (and the whole subtree under it) is moved still further down, and the severed right child of the node n's original left child (and the subtree under it) is made the left child of the node n in its new location. I.e., if we rotate the node 2 right, the initial situation of the tree depicted above transforms to: 
                             1
             4                               3
     8               2               6               7
16      17        9     5       12     13       14     15


Definition 4. The rotation of node n to left indicated here with ρn-1, is accomplished as above, but doing everything in its mirror image. This is the inverse operation to above. (Note the symmetry here, which differs from the asymmetry resulting from a slightly different definition of the binary tree rotation given in [1])


Definition 5. The rotation group of the infinite planar binary tree is the group generated by the enumerable set of the rotations ρn and their inverses ρn-1, with the index n ∈ N.


Theorem 1. This rotation group is isomorphic with the group A of the order preserving permutations of the rational numbers as defined by Cameron.
Proof. The abstract infinite planar binary tree defined above has its counterpart in the well-known Stern-Brocot tree which lists all the rationals once in their lowest terms, in such way that all the rationals that are located in the tree, to the left of any particular rational r are less than it in the usual magnitudewise sense.


Conjecture: The following infinite word in generators ρ1, ρ2, ρ3, ρ4, ρ5, ... where the generator ρn1 is the rotation of the nth branch of the positive Stern-Brocot tree (on the open interval ]0,∞[, where the branches are enumerated in breadth-first left-to-right fashion) to the right, and its inverse ρn-1 is correspondingly the rotation of the nth branch to the left, produces a permutation f : x -> x/2 in the positive half A+ of Cameron's group A of order preserving permutations of rational numbers.
The parentheses below are just for easier recognition of the pattern present:
(ρ11) (ρ21ρ30) (ρ41ρ50ρ6-1ρ71) (ρ81ρ90ρ10-1ρ111ρ120ρ13-1ρ141ρ150) ...
where the exponent en for the nth rotator-generator ρn is given by en = 1-((n-2[log2 n]) mod 3) [= A065251]


See the sequences A057114, A065249 and A065251 in EIS.
  1. Joan Lucas, Dominique Roelants van Baronaigien and Frank Ruskey, On Rotations and the Generation of Binary Trees, Journal of Algorithms, 15 (1993) 343-366.