covered is smaller than the total number of leaves you could have in a tree you just divide both sides by 2 to the L 1 and you get this thing over here so not too hard to prove the details of the proof don't matter too much but it can be done in one slide that's the first part how about the second part second part says for any set of lengths if this is satisfied then we can build a prefix third tree with those lengths how is this done you consider a full tree of depth L M which is the longest length and for each eye you pick any node of depth li