When breaking down a character, what do you do for parts where no radical exists? For example, 留. 刀 and 田 are obvious, but the top left? 匚 厶 come close but no match. Pleco says 卯, but I don't see this in many radical charts.
Actually if you can look at the traditional form of this character. The upper part is "丣".“丣” work as a phonetic radical here because it is pronounced you3. 

A lot of characters have been simplified in Mandarin in main land China. This make it harder for us to break down a character.

You can find more information here: 汉典解释 

So the first thing to do when breaking down a character is to find its traditional version.