  We consider the structure of a variation of the Fibonacci sequence which is
determined by a Bernoulli process. The associated structure of all Bernoulli
variations of the Fibonacci sequence can be represented by a directed binary
tree, which we denote X, with vertex labels representing the specific state of
the recurrence variation. Since X is a binary tree, we can consider the term of
a sequence variation given by a finite traversal of X represented by a binary
code t. We then prove that the traversal of X that is the reflection of the
digits of t gives exactly the integer term corresponding to t. We consider how
to further this result with the statement of an additional conjecture. Finally,
we give connections to Fibonacci expansions, the Stern-Brocot tree, and we
apply our methods to the Three Hat Problem as seen in ``Puzzle Corner'' of the
``Technology Review'' magazine.
