  Continuing our previous study \cite{Du:2011se} of permutation sum of color
ordered tree amplitudes of gluons, in this note, we prove the large-$z$
behavior of their cyclic sum and the combination of cyclic and permutation sums
under BCFW deformation. Unlike the permutation sum, the study of cyclic sum and
the combination of cyclic and permutation sums is much more difficult. By using
the generalized Bern-Carrasco-Johansson (BCJ) relation, we have proved the
boundary behavior of cyclic sum with nonadjacent BCFW deformation. The proof of
cyclic sum with adjacent BCFW deformation is a little bit simpler, where only
Kleiss-Kuijf (KK) relations are needed. Finally we have presented a new
observation for partial-ordered permutation sum and applied it to prove the
boundary behavior of combination sum with cyclic and permutation.
