Undertanding Aphids is the key to control. Since Aphids are more attracted to lush growth that is most often the result of using an excess of a Nitrogen fertilizer be sure the plant that is affected is fed a balanced diet.
Spraying any braod spectrum poison, that would be a pyrethrin based "insect control", will also kill off any predators that could aid in control, so the best means of control is a sharp stream of water to knock the wee buggers off the plant.

Here is a link that might be useful: About Aphids

One of the best methods of preventing a big outbreak of aphids in the early spring is to apply a couple of doses of horticultural oil along the stems, branches, and trunk of your tree. Aphids lay their eggs in the nooks and crannies in the fall, where they (eggs) overwinter safely. Oil applications can smother those fragile eggs, which are gas permeable. Until you spray them, of course, heh, heh, heh.

Pyrethrins are not effective (at all) to use as a preventative. And, as Kimmsr mentioned, a broad spectrum pesticide is rarely a long term solution. Such chemicals can often result in a boom in pest populations, following a reduction in the helpers.

Thanks for the reply, kimmsr and rhizo_1. First, I need to correct what I said in my original post. When I read your statements about "pyrethrins," I had to go look at what I had in the shed, because I know I got something different from described... What I REALLY have awaiting use is called "Pest Fighter, Year-Round Spray Oil." 98.8% **Paraffinic oil** Phew!

That said, I'm pretty sure the aphid problem was not because of too much N. Quite the opposite, I think. I'm fairly certain the plants out back weren't maintained/fertilized by the previous tenant--ever.

The issue we run into with continually spraying with water is that it isn't cheap here, and that task seems like it's fighting a losing battle instead of preventing one.

I read through the info kimmsr provided, and it states, "Usually natural enemy populations do not appear in significant numbers until aphids begin to be numerous." So, if I spray before the aphids become too rampant, will this help keep them under control without harming the beneficial guys?

Many times people think they need to use a hose and nozzle to spray off Aphids but a 1 quart bottle, or a 2-1/2 gallon sprayer, will work just as well and maybe better.

momma, your use of the paraffin oil has the potential of cutting way back on the spring outbreak of aphids. Use now and even once again before flowering.

Spraying now with the oil will do little to harm beneficial populations whatsoever. It won't completely prevent aphids, which love your kind of tree, but you should see a substantial decrease in numbers.