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In this paper, we consider a class of AND-OR tree circuits and study their response to random-pattern inputs as the depth of the tree is allowed to increase indefinitely. Each binary input of a circuit is independently chosen to be one (zero) with probability x (1 - x). The logic of the circuit determines the probability of success (one) at the output as a monotonically increasing S-shaped function of x called the probability transfer function. The probability transfer function of an AND-OR tree is shown to have just one interior fixed point (w.r.t. changes in depth of the tree) in the (0,1) range of x. Its value is of interest in random testing, being the input bias probability which optimizes the average length of random test for the circuit. The fixed point value is shown to be very sensitive to the fan-ins of the logic gates. As the depth of the tree becomes infinite, the probability transfer function becomes a unit step with the transition point located at the interior fixed point. We study the convergence to the unit step as a function of the circuit depth and the fan-in’s of the logic gates. The results are compared to other iteratively defined circuits whose building blocks also have an S-shaped transfer function.