  In graph theory, a tree is one of the more popular families of graphs with a
wide range of applications in computer science as well as many other related
fields. While there are several distance measures over the set of all trees, we
consider here the one which defines the so-called tree distance, defined by the
minimum number of edit operations, of removing and adding edges, in order to
change one tree into another. From a coding theoretic perspective, codes over
the tree distance are used for the correction of edge erasures and errors.
However, studying this distance measure is important for many other
applications that use trees and properties on their locality and the number of
neighbor trees. Under this paradigm, the largest size of code over trees with a
prescribed minimum tree distance is investigated. Upper bounds on these codes
as well as code constructions are presented. A significant part of our study is
dedicated to the problem of calculating the size of the ball of trees of a
given radius. These balls are not regular and thus we show that while the star
tree has asymptotically the smallest size of the ball, the maximum is achieved
for the path tree.
