If I am correct, the reason for string trees on certain guitars are because the strings must be at a certain angle (the break angle) to the nut/saddle. Does anyone know what this exact angle is? And what happens if this angle is not met?
Correct break angle is important at both ends.. not just the nut. The saddle must not be cut so low as to create a shallow break angle or this will make the string jump or vibrate in an unmusical way. This often happens on old acoustic guitars that have seen better days..someone fails to adjust the truss rod or the neck has moved and in order to lower the action, the saddle is lowered.. but you can only do this so much.. if a good break angle is lost the guitar looses good string tone and the string pressure on the saddle is lost.
The reason is actually to minimise the chance of the string popping out of the nut, and on some guitars to increase friction at the nut.
I have only had a guitar without a string tree cause problems once - it was a very cheap thing I had about 25 years ago, and when playing energetic rasgueado I could pop the D string out of it's notch on the nut.
Never had a problem since, and only two of my fixed bridge guitars have string trees.
Amusingly, one of my guitars with a tremolo and a locking nut also has a string tree - not sure exactly why...