AlphaGo does do a tree search as you say, but the "value" of the stopping node is half the neural network you mention, and another half performing Monte Carlo w

AlphaGo does do a tree search as you say, but the "value" of the stopping node is half the neural network you mention, and another half performing Monte Carlo with medium strength 'players' (2d amateur). These 'players' are really a quick version of the priority network that figures out which moves to play in the first place.