Distant parents in complete binary trees

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.