Time_Bound_in_Seconds may be omitted. In that case, there is no time bound on the testing and hence there is no time bound on the computation of the RH bound. We remark that computing the RH bound can take very long time (say, 1 day). This is why Time_Bound_in_Seconds can be helpful. No matter whether Time_Bound_in_Seconds is omitted or not, the programs always output the rMAAF bound for the trees in TreeFile. If Time_Bound_in_Seconds is 0, then the programs will stop immediately after computing the rMAAF bound. Otherwise, they will proceed to test if the RH bound is larger than i for i = L, L+1, ... (in this order), where L is the rMAAF bound.