You need only three pieces of information to create a fingering chart, including all alternate fingerings. Actually, once you understand the information, you won't need a fingering chart at all; you be able to construct the fingerings in your head anytime you need them.
Brass instruments have fundamental note on which the harmonic series is built. For a standard double horn, the fundamental for the F side is the C (written) below the bass clef, (sounding a fifth lower); for the B-flat side, it is the F (concert B-flat) just below the bass clef. A deskant (high F) horn has a fundamental an octave above the standard F horn. The fundamental of a natural horn depends on the key of the crook being used.
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The harmonic series starts on the fundamental pitch, then goes up in the pattern shown. These are the notes played on the open horn and on natural horns. The pitches above the fundamental are called overtones or partials. As you can see, the number of overtones doubles every octave as you go up. The octaves are always multiples of two (1, 2, 8, 16). The pitches with odd especially prime) numbers (7, 11, 13, for example) are the most out of tune and are to be avoided in most cases.
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The natural overtone series does not agree exactly with equal temperment, which is one of the reasons we have to adjust both within the instrument and with other instruments to be "in tune".
The fingering pattern can be started on any open tone. You can already see an alternate fingering -- G is an open note (i.e., an overtone), and also be played 1-3. You also see an indication of why notes above the treble clef are so difficult -- all those fingerings from the upper partials give the notes many alternate fingerings, so the embouchure has to be perfect to produce the right note.
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Now you understand the principles behind horn fingers, so you don't have to consult fingering charts any more.