  An added edge to a graph is called an inset edge. Predicting k inset edges
which minimize the average distance of a graph is known to be NP-Hard. However,
when k = 1 the complexity of the problem is polynomial. In this paper, some
tools for a precise analysis of the problem for the trees are established.
Using the tools, we can avoid using the distance matrix. This leads to more
efficient algorithms and a better analysis of the problem. Several applications
of the tools as well as a tight bound for the change of average distance when
an inset edge is added to a tree are presented.
