Scaling analysis combined with free-energy calculations and molecular dynamics simulations of a coarse-grained bead-spring model have been used to study the structural properties of planar brushes formed by bottlebrush polymers tethered by ends to a flat surface. We find that the height H of end-grafted bottlebrushes undergoes a sequence of conformational changes emerging from the overlap between molecules at subsequent length scales upon increasing the number of bottlebrushes per unit surface area σ. We establish a relation between H and the structural parameters of bottlebrush architecture: degree of polymerization of backbone Nbb and side chains Nsc as well as grafting density z of side chains. We find that nonoverlapping bottlebrushes exhibit the behavior of isolated polymers in a good solvent with the height that is σ-independent and scales as (1 + zNsc)2/5Nbb3/5. Weakly overlapping bottlebrushes are similar to end-grafted linear polymers (z = 0), and the height in this regime is σ1/3(1 + zNsc)2/3Nbb. Strongly overlapping bottlebrushes undergo a further increase in height, which is given by σ2/7z6/7Nbb.