What are they for? Why are they not necessary in some necks? Why are they bad for sustain? Is there any point in a frictionless nut if your guitar has string trees? How is a neck designed so that string trees aren't necessary?
Hmm, I am not sure where you are getting these ideas from but they are wrong...
On a fender headstock, Leo did away with the wider traditional slayed styles had and the typical angled back designs and difficult and often fragile scarf joint required to obtain it. With all six tuners in a row and no angle back, there is not enough pressure and some detrimental sympathetic vibration from the longer string lengths (high strings) and this is not a good thing.
what are they for...to increase the pressure over the nut and effectively shorten the length of string behind it that can interfere vibrate sympathetically with the main length of the string.
Why are they not necessary in some necks? a lot of necks have an angled back headstock to create the pressure required. Some now use 'staggered tuners' to create an effective back bend and pressure on the nut. Nuts like the LSR have tiny small pads behind the nut to stop sympathetic vibration.
Why are they bad for sustain? They are not, in fact in some cases they help by adding a bit of mass (though this is so small it does not matter) and to stop sympathetic vibrations. Not having enough pressure on the nut actually causes problems and especially with 'sustain' on open strings...so in many ways they can help...there is nothing about them that could make it worse?!
Is there any point in a frictionless nut if your guitar has string trees? A frictionless nut is a whole other thing. The string tress can often cause problems with friction, roller trees and such can help as well. A frictionless nut (though in reality there is no such thing, everything has some friction) is primarily so that the string runs smoothly over the nut and does not get hung up with trem use or tuning, even bending strings. Fenders also have the advantage of a straight string pull over the nut. Yes, in many cases, you may need string trees, even with a frictionless nut and staggered tuners.
How is a neck designed so that string trees aren't necessary? See above points and look at a lot of examples of guitars. A gibson might have as much as a 17 degree tilt on the headstock. If one addresses the sympathetic vibration and the pressure on the nut issues, it is possible to do without them. A frictionless nut will only help with things like tuning stability or because the material is better than the nut it replaces.
The point of a frictionless nut is that it's supposed to increase the sustain (wrong, it is there to prevent the strings getting caught up and affecting tuning stability).
Is there any point in that if you have string trees adding friction anyway? Obviously, for tuning stability and sympathetic vibration interference, you may address that through string trees as leo did...but adding friction is sometimes a by product. One cheap way to address things is to loosen off the string trees, perhaps so that they can 'rock' and yet provide the necessary down pressure and still dampen vibrations. Roller trees can also assist. Staggered tuners can provide the down pull usually, but do not address the sympathetic vibration aspect. An LSR can address the friction and have dampers, but not the down pull.
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So, there are a few examples of ways I address some of these things on my instruments...
Here is a typical fender head, the tuners are staggered and there is an LSR fitted...just behind the bearings are tiny foam 'dampers' that stop unwanted sympathetic vibrations. So, down pull, vibration and friction is addressed and string trees are not necessary and would only add friction...
Here is my novel solution for my LP with kahler trem. I used a graphtech 'tusq' nut and locking tuners that are un-staggered. No stagger is needed as you can see the head stock angles back substantially as is typical of such necks providing a lot of pressure over the nut. The tuners are 3+3 so there are no extreme length of strings to vibrate behind the strings either. I put together some rollers to create a 'straight through' pull over the nut and reduce friction over it from the side. The rollers do have a slight effect of dampening the strings behind the nut as well and the plate it sits on adds a fair amount of 'mass' to the headstock to help dampen it from sympathetically vibrating (You can play with this idea, hold your headstock against something solid to stop it vibrating as an experiment)...plus, it strengthens that notorious weak spot on such necks where you have a scarf join at the thinnest part of the neck with a great big chunk out of it for the truss rod adjustment.
Here is a pic of my strat. Again, like the tele staggered tuners. The headstock is a bit 'deeper' so there is even more pressure on the nut. I did not have a problem with the stock nut so it is nothing 'exotic'. However, I did have a problem with sympathetic vibration and evenness with the open string so the string trees add that little extra and cut the length of string into smaller sections on the 4 higher strings. You can see that the angle down off the back of the nut is fairly even across all string where it would be a lot shallower if not for those trees, despite the staggered locking tuners.
Despite the nut not being 'friction-less' it stays in tune remarkably well as is with good nut slots and straight through string pull, locking tuners and a nice pressure on it (not too much as many traditional trees can do and should be 'lifted' and be able to move with the string...or better yet replaced with rollers of graphite versions).
The tele is by far the best for tuning with a fully floating wide range trem it can emerge from a long drive from a soft case with the trem depressed and spring back virtually in tune without any adjustment.
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I'd be interested to know where you got those ideas from as we do like to clarify some of the mojo about around here and perhaps it is 'advertising' speak that leads to the notion that a frictionless nut has anything to do with "sustain" per se.
Regardless, I hope that gives you some answers or ideas ~ p