Let $A=\{1,2\}$. For any $d \in A$ and any sequence $a=(a_1,a_2,\dots)\in A^{\mathbb N}$ the associated rooted tree $T(d,a)$ is recursively defined in the following way. The degree of the root of $T(d,a)$ is $d$, and the result of the removal of its root (together with the $d$ edges issued from it) is the tree $T(a_1,Sa)$ if $d=1$ and two trees $T(1,Sa), T(2,Sa)$ if $d=2$ (here $Sa=(a_2,a_3,\dots)$ denotes the shift of the sequence $a$).
I am interested in any references to the appearances of the family of trees $T(d,a)$ (or anything similar).
