The answer is quite simple.
A sound wave is generated, if by "sound wave" you mean a wave of air emanating because of the falling tree.
However, a sound is only a sound if there is some receptor there to translate the air wave into sound.
I'm not sure I buy the distinction made between a "sound" and a "sound wave."
The sound is the sound wave. Otherwise, it's like saying that a beam of light isn't really light until an eye (or other receptor) sees it. The receptor has no cause in the thing that causes the sound or the light. The receptor's detection of the sound or the light doesn't somehow change the nature of the sound in terms of what it is. It only observes it.
If something exists, it doesn't exist only because an observer observed it. It exists regardless of whether anything or anyone observed it.
A "sound wave" is a misnomer actually. It is really only an air wave.
An air wave needs a receptor that can translate the change in pressure on the receptor into a phenomena we interpret as sound.
In space, despite almost universal depictions to the contrary, if something blows up, you won't hear it unless there is some substance that results in changed pressure on your ear drum.
Or, a deaf person doesn't hear your voice, even though your vocal cords vibrate the air around them. Why not? Because the deaf person's receptors are broken.
Ergo, air movement, but no sound.