I read the question as referring to total radiation and not a specific
angle that you referred to.
With respect to a single reflector it is obviouslythe most significant
addition to a dipole (efficiency
addition if you like)and additional elements added follow a declining
level of addition ofredirected energy ( binomial) which I took as his
question. The other questions he raised were follow ups on the yagi
design which fits nicely to the vector style explanation without going
to deep into true antenna radiation thus raising the question of
reflector "length" It really is the same as explaning
"skin depth" without reference toi "volume" or capacitance that does
not refer to "skin depth",.
Most who ask a question may well accept "thats the way it is" but the
gentleman asked the question with respect to reflector length because
of yagi type explanations, which does not "fit"
with all arrays which are actually meshed cuircuits and more difficult
to address.
I have no quarrel with the choice of a two element antenna in the real
world because as Moxon states it is simpler and more rewarding to raise
a two element antenna than to make a three element antenna BUT
it depends on the end parameters that you require which if not stated
lead to perennial radio antenna arguements which plague forums.
Nuff said Art