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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.

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I guess a different way of asking this question is whether or not all parts of a character have a radical. Also, why is this info useful? – xiaohouzi79 Nov 20 '13 at 0:19
As a matter of fact, some characters use their first stroke as a radical in the Xinhua (New Chinese) Dictionary. Besides, there's usually an index ordered by the number of strokes to facilitate the lookup process of a hard-to-break-down character. – phoeagon Nov 20 '13 at 4:56

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.

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