I've tried with a small hoe and got my self in a world of trouble, if the tree ever wants to come back (wind, whatever), it will quickly overpower a small machine.
I've done a number of trees with an excavator (25,000 lb machine) and the trees were oaks about 40 - 50 foot high, the excavator can get about half way up the tree to push.
I've never felled a tree, other than very small, in a direction other than the one it was leaning towards.
I'll dig out the roots on the oposite side and then push the tree over with the excavator.
When in doubt with a tree I'll
cut it down and then deal with the stump, I'd much rather be running away than trying to climb out of a machine if something were to go wrong.
Then again can always hire an expert if the tree is in a really tough spot.
Oaks at my house are nasty to get out, deep tap root, and they weigh a ton, they'll grab every rock within 10 foot of them.
Joel