When building a chord like “C13th,” there are seven notes shown. The most notes that can be covered by one hand on the fretboard at one time is six. So, which note (s) do you leave out in order to actually play the chord?
As you can imagine, there's a lot of debate amongst music theorists as to what is the “proper” thing to do in such a situation when you have a chord that has more notes than you are able to produce.
Traditionally, the fifth or the third would be left out (usually in that order). Believe it or not, there are instances, though, when the root is the “missing” note.
But the real determining factor is what notes you are able to finger (or not finger) on your fretboard. For instance, if you strum your guitar (standard tuning) without putting any fingers on the fretboard at all you would have an A11. The notes, from low to high, would be E (fifth), A (root), D (eleventh), G (seventh), B (ninth) and E (fifth again). Here the third (C#) is the missing note. You could always add this by playing it on the 1st (or 6th) string but it sounds perfectly fine as it is.
Generally a good rule of thumb with 9th, 11th, and 13th chords is to really try to include the seventh along with the root in order to give it some sense of identity.