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We investigate the phase diagram of branching annihilating random walks withone and two offsprings in one dimension. A walker can hop to a nearest neighborsite or branch with one or two offsprings with relative ratio. Two walkersannihilate immediately when they meet. In general, this model exhibits acontinuous phase transition from an active state into the absorbing state(vacuum) at a finite hopping probability. We map out the phase diagram by MonteCarlo simulations which shows a reentrant phase transition from vacuum to anactive state and finally into vacuum again as the relative rate of thetwo-offspring branching process increases. This reentrant property apparentlycontradicts the conventional wisdom that increasing the number of offspringswill tend to make the system more active. We show that the reentrant propertyis due to the static reflection symmetry of two-offspring branching processesand the conventional wisdom is recovered when the dynamic reflection symmetryis introduced instead of the static one.Comment: 14 pages, Revtex, 4 figures (one PS figure file upon request)  (submitted to Phy. Rev. E

Reentrant phase diagram of branching annihilating random walks with one and two offspring