Read this a long time ago, as a child. Perhaps did not get the full structure. . .
it opens with Aerin the king's daughter, but not heir, musing about her mother, and where she had first heard the story about her mother being a witch-woman from the North who had enchanted her mother. Not from her cousin Galanna, despite all the other trouble she's been. . . . And then she asks her father for leave to come with him on a mission, and her father, and her father's heir and her cousin Tor, must reject it, and Perlith, third in rank, bitterly ridicules her asking.
We go back in time for why this is so -- and why Perlith bitterly refers to her as a Dragonslayer -- for Part I. It involves a lame horse, Aerin's making an invalid of herself for a long time, reading a book, buying herbs, a ballad about her red hair, and much more.
And in Part II, why being the Dragonslayer, and the secrets of her mother, are so important, are brought forth in a plot that involves wild cats and wild dogs, climbing a stair for a long time, a jungle that has no business being there, a lake, and a dragon skull.