This is a good question. I read A Tree Grows in Brooklyn about two years ago for the first time. It does contain mature material.
While some children may mature faster than others, I erred on the side of caution with my own children. In my own opinion, I would not recommend A Tree Grows in Brooklyn for a ninth grader regardless of the child's maturity level. However, the beginning of the novel has some excellent passages that you may wish to share with your daughter because many of these early passages are appropriate relating to the main character's struggles. You could read these passages aloud and discuss them. The latter part of the book deals with dramatic adult matters that your daughter doesn't have a context for.
There is plenty of time for her to read this book in perhaps twelfth grade. I was glad that I didn't read it until I was an adult because I could put all of the character's life events in perspective. As I remember, one of the passages was very disturbing to me and the image stayed with me and unsettled me. You don't know what might unnecessarily upset your daughter. While this story is a well-written classic, it probably isn't worth disturbing your child's peace and perhaps stirring up fears.
Peace be with you,