Unless there is a Form 8332 filed with the non-custodial parents taxes the custodial parent is given the ability to claim the exemption.  This is the rule regardless of how long the child lived with the parents together, or even the non-custodial parent.  For example, if the child lives with the father for 7 months, but then moves to the mother’s residence the mother is entitled to claim the exemption since that is the parent with custody at the end of the year.