Laziness invariance of branching extinction ($G_q(z)=qz+(1-q)G(z),q<1$)

For a Galton-Watson process, replacing an individual's reproduction by a one-descendant uneventful step with probability $q<1$ changes its offspring probability generating function to the displayed mixture. Its fixed-point equation reduces to $G(z)=z$, so the smallest fixed point and hence the extinction probability of a branching process are unchanged. The mean is $q+(1-q)G^{\prime }(1)$, which also preserves the subcritical/critical/supercritical classification. The excluded case $q=1$ never reproduces or dies and need not share the extinction probability.