The clerk of the court in which such judgment is rendered shall certify a copy of the assessment roll as confirmed, and of the judgment confirming the same, to the treasurer of the county, or if there has been an appeal taken from any part of such judgment, then he or she shall certify such part of the roll and judgment as is not included in such appeal, and the remainder when final judgment is entered: PROVIDED, That if upon such appeal the judgment of the superior court shall be affirmed, the assessments on such property as to which appeal has been taken shall bear interest at the same rate and from the same date which other assessments not paid within the time hereafter provided shall bear. Such copy of the assessment roll shall be sufficient warrant to the county treasurer to collect the assessments therein specified in the manner hereinafter provided.