As well as a carbohydrate source, yeast theoretically need some ammonium, various B vitamins, sulphur, phosphorous, magnesium, copper, iron, and zinc (see Fix's brewing science book). It is interesting to compare these theoretical requirements with actual practice. There is widespread consensus from those who do all-sugar brews that a combination of di-ammonium phosphate (DAP), Vit B complex and MgSO4 will result in a perfectly healthy ferment. From this, it might be concluded that all the other minerals are provided by the water supply and/or the boiler construction, with an obvious test being to use distilled water and a plastic boiler. On the other hand, it is unlikely that enough sulphur comes from the water, leading to the possibility that the pitching yeast is also a source of some nutrients. (I must admit at this point to being a bit hazy as to what minerals might be in a Vit B complex - I know cobalt is one). One difficulty with this minimum nutrient concept is that yeast are literally chemical factories, and so we need to distinguish between what they absolutely can't do without and what they can synthesise under duress, possibly with delays to the ferment and a change in the byproducts profile.
One takeaway message is that vodka brewers generally don't feel the need to use Zn supplements, and in theory at least that ought to go double for beer brewers.
Many people use old yeast as nutrient, but apparently it is better to use the correct nutrient rather than dead yeasts. An analogy was, would humans survive and thrive just eating dead humans?
A counter-analogy is whether you think people would do better eating real organic food or some minimalist inorganic combo that was cooked up in a lab, based upon imperfect ideas about nutrition. Yeast contains everything that yeast needs, and one plus could be that the minerals are appropriately chelated. One reservation could be that both the amino acid and the saturated lipid content could lead to more fusels, which I guess could be regarded as either a good or a bad thing depending what style you're shooting for. But the same considerations apply to live yeast pitching rates.
In theory, 4g/L of dead yeast provides about the same nitrogen as 1g/L of DAP. In practice, I must admit that the DAP results in a faster ferment, although the yeast-based nutrient is perfectly adequate for an all-sugar SG 60 brew. Increase rates appropriately for higher gravities, but decrease rates appropriately for beer worts as these will always attenuate at least 50% without nutrient assistance.
What about boiling up some trub for previous brews as a nutrient?
Do you mean break or ferment dregs? The advantage of the former is that you get all those unsaturated lipids which can substitute for wort oxygenation, plus a heap of precipitated protein (ie, an amino acid source). However, I am unsure whether trub in isolation is particularly rich in Vit B. The ferment dregs would contain the lot.