When the tones C, E and G are played, C is perceived as the root. In the case of this major triad, the root is easily identifiable. However, for many chords, there is no identifiable root. For example, the tones C, D and E do not indicate any tone as the root tone.
The ways to identify the root of a chord are fairly well understood. Whenever a fifth interval is present, the bottom tone must be the root. (For example C,F -> F is root). Whenever minor or major third intervals are stacked, the bottom most tone is the root (For example A,C,D,F -> D is root).
Is there any mathematical / physical theory that describes why chords have roots? It would seem that based on the general qualitative rules I listed above the root tone mechanism could be deduced. I have looked into studying the wave interference and frequency ratios of intervals that make up chords but so far I haven't found an obvious answer.
-- EDIT --
I am adding an edit here to make clear the definitions I am using. I know different people have different ideas about what a "chord" and "root" really are, so I want to be as specific as possible:
Chord - A set of unique tones. (For example; CEG, DFA, ABC, ABCDEFG,...)
Root - One or more tones in the chord that are perceived as the "net" tone. The root tone(s) is therefore used for analyzing the tonality between two chords, like a chord progression. (For example: CEG -> Root is C, C E G# -> Root is C E G#)