The breeder's Animal husbandry affects the minimum and maximum number of traits that will appear in a horse. The maximum number of traits you can breed into one animal is currently 8 at 80+ skill.
At lower levels, breeding animals with traits will generally result in around AH skill/10 amount of traits. i.e. 5 traits + 5 traits with 30 AH will most often result in 3 traits for the new animal, however there is some level of RNG and it is possible to obtain 4 and 5 traits from breeding at this level.