  We study the random Fibonacci tree, which is an infinite binary tree with
non-negative integers at each node. The root consists of the number 1 with a
single child, also the number 1. We define the tree recursively in the
following way: if x is the parent of y, then y has two children, namely |x-y|
and x+y. This tree was studied by Benoit Rittaud who proved that any pair of
integers a,b that are coprime occur as a parent-child pair infinitely often. We
extend his results by determining the probability that a random infinite walk
in this tree contains exactly one pair (1,1), that being at the root of the
tree. Also, we give tight upper and lower bounds on the number of occurrences
of any specific coprime pair (a,b) at any given fixed depth in the tree.
