Dear Rose,
Please, I understand your concern yet since your daughter is only a year old, I would refrain from worrying so much that your worry rubs off on her or you get overly paranoid. There is a slight chance she will develop SZ yet as you already suspect, the quicker you got her treatment, the better the prognosis would be.
I suggest you Google "schizophrenia early warning signs," as well as keep reading the Schizophrenia information under the Find menu, under "Schizophrenia Basics" on this Connection web site. Educate yourself so that when you daughter is a teenager, you're alerted to anything out of the ordinary.
For now, I will tell you what I tell every mother or father to do as a measure of protection: The so-called "pre-morbid" functioning determines how well a person fares after she develops SZ. So, early on in her life, I suggest you get her involved with other children her age, take her to do things outside the house. Then, as she gets older, encourage her to do things like volunteer or work at a part-time job, when she's a teen. The more she's able to function, the better her chances will be should she develop SZ.
I would relax, enjoy your daughter as she grows up, and one last thing: read parenting books as I'm not an expert on babies and toddlers and children. My main concern is that you could "read into" her behaviors and worry too much.
Most of all, show and tell her you love her, often.
By the way, I was diagnosed with SZ when I was 22, I'm 43 going on 44, and I recovered from the SZ, so there is hope.
Best regards,
Christina