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 sigma. We establish a relation between H and the structural parameters of bottlebrush architecture: degree of polymerization of backbone N-bb and side chains N-sc 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 sigma-independent and scales as proportional to (1 + zN(sc))N-2/5(bb)3/5. Weakly overlapping bottlebrushes are similar to end-grafted linear polymers (z = 0), and the height in this regime is proportional to sigma(1/3) (1 + zN(sc))N-2/3(bb). Strongly overlapping bottlebrushes undergo a further increase in height, which is given by proportional to sigma(2/7)z(6/7) N-bb.