  There is a unique path from the root of a tree to any other vertex. Every
vertex, except the root, has a parent: the adjoining vertex on this unique
path. This is the conventional definition of the parent vertex. For complete
binary trees, however, we show that it is useful to define another parent
vertex, called a \emph{distant parent}. The study of distant parents leads to
novel connections with dyadic rational numbers. Moreover, we apply the concepts
of close and distant parent vertices to deduce an apparently new sense in which
continued fractions are `best' rational approximations.
