Whenever only a portion of a parcel, tract, or lot of real property which was previously taxable becomes exempt from the levy and collection of property tax for any reason, the treasurer may, upon the basis of an appraisal and computation of the valuation for assessment of such property by the assessor, either collect the property taxes thereon for the current taxable year, calculated on the basis of the property tax levy on such property during the preceding taxable year and prorated to the date upon which title to such property was conveyed, or, if the treasurer is satisfied that there is sufficient taxable real property remaining to satisfy any lien for the amount of property taxes payable on such portion, he may defer collection of the property taxes until the following taxable year. In the event the prorated taxes on such portion are collected, the owner of the remainder of such real property shall be credited with the full amount of taxes collected when the property tax levy for the current taxable year has been fixed and made and the correct amount of property taxes determined.