# What would happen if we added this rule to red-black trees?

So, I know that a normal r-b tree has a height of O(logn). What would happen is we let a red node have a red child if its parent is black?

Would the height still be O(logn)? Would you have to have a totally different insertion scheme to maintain the height?

Yes it stays $$O(\log n)$$. Compared to the proof of the original one, we have no three red consecutive on a path from the root to a leaf. The bound on the blacks stays the same in a path. The length of the path is now at most three times the number of blacks which is at most $$3O(\log n)=O(\log n)$$.