Here is a little explanation of how the Boltzmann and Saha equations
behave as a function of their input parameters.  

The BOLTZMANN EQUATION gives the ratio of the number of atoms in excitation
state B N_B to those in excitation state A N_A, where the 
energy_level(B)>energy_level(A).

1) If we look at how the relative populations change with FIXED TEMPERATURE
and VARY THE ENERGY DIFFERENCES, we see that the ratio quickly drops as
the energy difference between the two levels goes up.  This makes sense,
because for a given temperature, fewer and fewer particles will have
high enough kinetic energies (=velocities) to make collisions to 
produce big energy jumps for electrons.  NOTE THAT ONLY DISCRETE ENERGY
DIFFERENCES ARE ALLOWED BECAUSE ENERGY LEVELS ARE QUANTIZED FOR BOUND
ELECTRONS!

2) If we look at how the relative populations change with FIXED 
ENERGY DIFFERENCES and VARY THE TEMPERATURE, we see that as the temperature
goes UP, the ratio N_B/N_A (normalized by the ratio of statistical weights)
approaches one. That is, more and more atoms are in the higher
energy state.  At low temperatures, almost all the atoms are in the
lower energy state, because almost no collisions are energetic enough to
excite an atom into state B.  At high temperatures, just about every
collision will do that (or more).  However, as electrons get excited, they
quickly de-excite/cascade down, with a given probability.  So, even if
all the atoms are instantaneously in state B, they will quickly go back
to state A.  How fast they go back depends on quantum mechanical
probabilities -- which define the statistical weights.  So, at any given
time, the largest that N_B/N_A can be is the ratio of the statistical
weights.  Remember that if temperatures are really high, electrons can
be excited to ever high levels, so state B might not be the dominant state.
Then both states A and B may be "temporary" states as electrons constantly
are excited and cascade down.  Again, N_B/N_A in this case will be 
dicated by the ratio of the statistical weights of those probabilities.
Note that two difference kT/Delta_E regimes are plotted, one 0 to 5, 
and the other 0 to 30.  You can see how quickly N_B/N_A asymptotically
approaches the statistical weight ratio.

The SAHA EQUATION tells us the ratio of atoms in ionization state i+1
compared to ionization state i, or N_(i+1)/N_(i).  
The behavior of the two main factors
and their product are plotted for two ranges of kT/chi_0, from 0 to 2.5
and for 0 to 15.  You see that the decaying exponential approaches unity,
but not terribly fast.  For x=chi/kT=1.0,2.5,5,10.,20.
exp(-1/x)= 0.37,0.67,0.82,0.90,0.95.  At low temperatures (or
high ionization potentials), hardly any ions are in the higher energy
state because hardly any collisions are energetic enough to 
ionize atoms from state i to i+1.  As the temperature goes up
(or for smaller ion energy differences), more and more collisions
are able to ionize atoms from state i to i+1.  The trend continues
and the ratio asymptotically approaches the (kT)^3/2 curve.
This means that N_(i+1)/N_(i) *KEEPS INCREASING* with temperature --
even if most of the ions might be in N_(i+2),N_(i+3) etc.
The product (x^1.5)(exp(-1/x))~0.95 for kT~1.5*chi_0 (1.5x the
ionization potential).  Note that for kT~chi_0, the (x^1.5)(exp(-1/x))~0.37
(1/e).  We also have to remember that there is a coefficient
A which is on the order of 10^56 (mks units) for ionized to neutral
H (N_+/N_0), and that we divide by the number density of electrons.

For H, it's easy to see that the neutral fraction just gets smaller and
smaller with increasing temperature.  That makes sense.  What about a
multi-electron atom like O?  There are 9 possible ionization states,
from O I to O IX.  If we consider pure O gas and think about what happens
as temperature T increases, there is not a lot of increase of O II/O I
until kT ~ 1.5*chi_0 (1.5x the ionization energy for the neutral state).  As
T continues to increase, N_II/N_I (=N_+/N_0, adopting the astronomy
usage that O I is neutral, O II is singly ionized etc.) 
will grow like (kT)^3/2. At the same
time, N_III/N_II(=N_++/N_+) will be small, but will start to rise as
kT approaches 1.5*chi_1.  By that point, N_II/N_I will be really high -- 
but N_III/N_II will be substantially greater than unity, so
N_III/N_I will be REALLY high.  The progression will go on.
By the time N_VI/N_V is on the order of unity, the number
of O atoms in states O I, O II and O III will be negligible.
At the same time, N_VII/N_VI will be <1,, N_VIII/N_VII <<1 etc.
So you see that there are going to be very few atoms in ionization states
more than one ionization level away from the "dominant" ionization
state for which kT~chi_0.  This justifies 
N_i,s ~ N_i,s/(N_i-1 + N_i + N_i+1)  equation (8-33)