Is it obligatory for children to live with their father if he treats them well but the their mother is not present in the home?

The mother has the strongest right to the custody of her un-discerning child, by consensus, unless there is a impediment. As for a female child after reaching the age of discernment, scholars have differed. Therefore, it is best for you to obey your father by residing with him as long as he is fit for custody and treats you well, while continuously maintaining ties with your mother and being dutiful to her.