The classic analysis of the probability that the Galton-Watson tree is infinite gives an equation with two solutions. What about other properties. For first order properties we show that the equation system has a unique solution. More generally we consider properties defined by tree automata. There is then a natural equation system. Sometimes the system has rogue solutions, meaning solutions with no interpretation. This area combines structural combinatorics, probabilistic combinatorics and logic, all topics Jirka loved. Joint work with Moumanti Podder.