Positivism is reductionist theory, meaning that it proposes that all valid information is derived from logic, mathematics, and the scientific method. In other words, all knowledge can be explained (reduced to) laws derived from strict scientific practice. Auguste Comte would popularize positivism by using it to study sociology. He supposed that societies adhere to certain laws in the same way that a physical phenomenon in the universe does.
Being a strict scientific theory, a positivist might say that since we cannot verify the sound the tree makes (since we are not there); then it can not be considered valid knowledge (i. e. true or meaningfully true). Still, being a theory based on scientific laws, if we knew all the laws governing sound waves, air vibrations, and so on, the positivist might conclude that the falling tree does make sounds whether or not someone is there to hear them. Let's say that based on all other experiments with observers, the falling tree made sounds 100 out of 100 times. Therefore, it seems like a proven hypothesis that it is extremely likely that the tree will make a sound regardless of whether or not someone is there to hear it.
But if the positivist is quite strict in his/her positivism, he/she would reduce the laws to laws of physics. Then it becomes a question if the propagation of sound waves is the same thing as sound itself; in other words, are vibrations of the air ("perceivable phenomena") and their reception in the ear ("perceived phenomena") the same thing . . . scientifically? The strict positivist would say that they are not the same thing. A deaf person, standing near the tree as it falls, might feel the vibration on the ground, but he would not hear the "sound" of the tree falling even though, clearly there were sound waves propagating through the air. Considering this, sound waves and "sound" are not the same thing. So, in that respect, there would be sound waves, but with no one there to hear it, let alone be able to hear it, there would be no "sound."