The invention relates to a tyre for a private passenger vehicle comprising a tread extended on either side by flanks, that tread being provided with an asymmetrical sculpted design relative to a median plane dividing that tread into two parts with a same axial width, that axial width being measured in a direction parallel to the axis of rotation, an outer part and an inner part, that outer part being situated between that median plane and the flank designed to be placed on the outside of a vehicle and the inner part being situated between the same plane and the flank designed to be placed toward the inside of the same vehicle, the inner part and the outer part each comprising at least one hollow forming a slot or designed to form a slot from a partial wearing-off of the tread, those hollows defining, in new condition and for each part (inner and outer), a total hollow volume, ViO and VeO, respectively, a hollow surface rate, TiO and TeO, respectively, those rates being different from zero and such that in new condition, there is a ratio RO between the hollow volume of the inner part and the hollow volume of the outer part, and in each worn condition (U) chosen between the initial state and the wear limit of the tread, there is a ratio RU between the hollow volume of the inner part and the hollow volume of the outer part, said tyre being characterised in that, at least from 80% wear measured on the tread thickness to be worn off, the ratio RU is greater than the ratio RO.