  We define and study a new compactification, called the height
compactification of the horospheric product of two infinite trees. We will
provide a complete description of this compactification. In particular, we show
that this compactification is isomorphic to the Busemann compactification when
all the vertices of both trees have degrees of at least three, which also leads
to a precise description of the Busemann functions in terms of the points in
the geometric compactification of each tree. We will discuss an application to
the asymptotic behavior of integrable ergodic cocycles with values in the
isometry group of such horospheric product.
