The tritone is the definitive constituent of a Dominant 7 chord (a major triad with a minor 7 causing a tritone between the 3rd and 7th of a chord. A dominant 7 chord can be defined just by its tritone. In other words, a tritone, without any other notes, is indicating that the root is a major third below one of the two notes.
For example, the tritone E - Bb: A major third below E is C defining a C7, a major third below Bb is Gb defining a Gb7 (the E now being spelled as Fb-- these two 7ths share the same tritone and each represents the tritone substitute for the other. This is why there are often proscriptions about adding tensions (9, 11, 13 and their alterations) that cause tritones between upper members of a chord -- they cause ambiguity. The tritone tries to indicate that the actual root is the note a major 3rd below one of its members. Try this. Start with a tritone G# D, move down a half step (G C#), another (F# C), one more (F B) and finish with E C interval. Now, to each one add a bass moving in 5ths E/ G# D, A / G C#, D /F# C, G / F B, C / E C. Now try it with the bass moving by half steps from E (E, Eb, D, Db, C) then from Bb (Bb, A, Ab, G, C -- note the final 5th movement.) The progressions work either way and can ultimately move to the same place (C).
Example -- the C maj 7 usually will not contain the natural 11th (F) because it contains the tritone B - F: the prime constituent of a G7 chord. If we look at the two C maj 11: C E G B D F A and G11: G B D F A C, they contain the same pitches. It wants to be a G11 but it has the "wrong Bass."
This idea is generally true for both major and minor chords. Of course, the m7b5 (half-diminished) and the diminished 7 both contain tritones. Both of these often behave as a V chord (9 or b9) without a root (a kind of disguised V).
The only chord that may be said to be somewhat minor in quality and contains a tritone is the minor 7 b5 when used as a form of II in a II - V progression. This chord comes from Harmonic and natural minors and acts as an altered minor 7. In this case, the tritone is between the root and the b5.
The diminished 7 chord contains 2 tritones and as such can act as a rootless b9 harmony over one of four roots. (A C dim == C Eb Gb Bbb(or A) thus its potential roots are: Ab, Cb(B), D(Gb = F#), or F)
Adding a minor chord above a tritone will generally crate a rootless voicing of some dominant 7 chord. Those that work can be explained and have a tendancy to act as a dominant based off of a maj 3rd below one of the tritone members.
F B / D F A (Dm) = G9 (rootless); F B / Ab Cb Eb (Abm) = G7 #5 b9 (Eb = D#, Cb = B). Many of these, the root can be changed to its tritone substitute (you get two dominant voicings for the price of one).
Rootlessness is part of the reason for voicings with the tritone as the lowest part. It keeps the keyboard player out of the way of the bass player allowing that person freedom.
There may be instances out there that context will allow the feeling of minorness over a tritone. But these are rare at best (I've been playing for 40 years and can't think of one). But, if you realize that in general, a tritone represents a dominant 7 type chord with the root being a major third below one of its constituents. When ever you see any dominant 7 chord, you can represent it with just the tritone.
