I would think a regular 30 inch bow saw and a pair of loppers would take care of it. That is what i use if it's six inches in diameter or less. For me not worth the time to crank up my chain saw for something that small.
If you can post some pics. someone may have a better idea.

Just a follow up, I am attaching a picture of the tree in three different posts. Upon closer examination it is a bit larger than in my original post and probably15', but the trunks are not that thick and there is no real danger of damaging property. I included some plastic cups to give you an idea of the scale of the tree.

2nd of three pictures of the river birch base.

last picture, (3rd of 3).

last picture, (3rd of 3).

{{!gwi}}

If I were to cut that tree down I'd use a sawzall type saber saw with a quality 12 inch pruning blade. It's almost as efficient as an electric chain saw but easier to control. Less likely to grab or throw something harsh because it reciprocates rather than spins.

I'd then cut it flush with the ground using the saber saw. Just lay the saw on its side and slice along the dirt. Removing the dirt somewhat beforehand with a hose with a jet type sweeper nozzle would maybe help to get the wood more exposed to help avoid trying to slice through flint rocks. At the most, I'd destroy a ten dollar blade; nothing more would be needed in equipment or disposables.

Then I'd cover it with manure or fertilizer (nitrogen sources) and a low mound of soil, and the little stump would be gone in a year or two, depending on how moist it was kept. Some people say this doesn't work, but it does for me. Reliably.

BTW: Harbor Fright has a 6 amp recip saw on sale right now (Item #65570) for $29, and that's a nearly throwaway price.

Great advice, thanks so much.

I really didn't want to get a chainsaw because of control issues (I"m a "mature" woman) and I have a childhood memory of my father injuring himself with a chain saw...so your reciprocating saw suggestion is preferable because it seems safer and I can easily see other practical uses for it.

If I could add my $.02 to this....Great advice on cutting it down from grubby_me. There is a product, and the name escapes me at the moment, that accelerates the decay of the stump. Once you get the tree cut and the stump leveled with the ground, make several cross-cuts in the stump. If you aren't worried about adjacent plants, pour some Round-up on the stump to ensure no additional leafing out occurs (you can skip this if you prefer). The product I'm talking about to accelerate decay is a powder you sprinkle on the raw stump. The cross-cuts should ensure good penetration. Then cover with dirt and mulch and you should see these stumps gone.