Prove that every thin AVL tree may be converted to red-black tree
Just a remark.

Your "thin" trees are sometimes know as Fibonacci trees. A Fibonacci tree $T_h$ of height $h$ is recursively defined as root with two subtrees $T_{h-1}$ and $T_{h-2}$. Hence the name.

As the number of black nodes along each path from root to leaf must be equal, this means that the subtrees $T_{h-1}$ and $T_{h-2}$ must have the same black height  in $T_h$, although they differ in height.