Wanna do more pull ups?
Contrary to what you might think, soley working on doing many reps of pull-ups is not the optimal way to increase the maximum amount of pull ups you can do.
Endurance is actually related to strength in that you have a very high endurance when lifing loads lighter than 25% of your 1-repetition maximum. Therefore, increasing your 1-RM, i.e. increasing your strength will help you immensely with increasing the maximum amount of reps you can perform with lighter weights.
Therefore you should train pull-ups with heavy weight or do heavy lat pull-downs with a load of 1-4 RM. Of course, high rep training is also beneficial, especially for increase light weight endurance. Therefore, ideally, a combination of both training methods will allow you to maximize your pull-up reps. .
This is based on comparisons between wrestlers and weightlifters where wrestlers can do a few more more repetitions for any given % of their 1RM (they have high muscular enduance). The benefit is not that high for heavy weights around 60% of 1RM.
Weightlifters, however, are generally stronger and will be able to do more repetitions of a given relatively heavy weight.
Vladimir Zatsiorsky, William Kraemer. Science and Practice of Strength Training. January 2006.