  Using explicit methods, we provide an upper bound to the norm of the Busemann
cocycle of a locally finite regular tree $X$, emphasizing the symmetries of the
cocycle. The latter takes value into a submodule of square summable functions
on the edges of $X$, which corresponds the Steinberg representation for rank
one groups acting on their Bruhat-Tits tree. The norm of the Busemann cocycle
is asymptotically linear with respect to square root of the distance between
any two vertices. Independently, Gournay and Jolissaint proved an exact formula
for harmonic 1-cocycles covering the present case.
