Agree with Rich in regards to rain and the softness of the ground. I can tell a HUGE difference knocking down tress when our clay soil is wet/moist vs. dry as a brick here in NC.
I have a 45HP 4WD Jinma and typically I just knock the trees down. Go in with the bucket about 4 or 5 ft high and just start pushing. When the tree starts to go over I lower the bucket, while continuing to move forward, providing more leverage which will typically bring up most of the roots. This works pretty well on anything up to about 6" dia and some trees up to about 10" that do not have a long tap root.
When the ground is really wet I have knocked over 10" to 12" dia. Usually go higher up on the trunk, but, have to be careful that the root ball does not come up under the tractor when going higher up the trunk. Read once here on TBN where a guy got his front end off the ground when the root ball popped out and was under the tractor. Took a long time to get the tractor off the root. Once I feel the tree starting to go over, I will stop moving forward and just lower the bucket.
Have tried yanking out some of the more stubborn roots, but, have not had a lot of success. Have dug out around them with the FEL, do not have a BH, tied a chain to them and started pulling but usually just end up spinning. I had a particularly stubborn pine root that was about 16" dia that I just could not get out. After spending the better part of a couple of hours digging a "pit" around three sides of it, I was able to push it out by backing into with the back of my box blade.
The type of root makes a difference as well. Trees with large tap roots, mostly hardwoods & "nut" trees (Oaks, Elms, Maples, etc.) will be a little harder to remove. Trees with a fibrous root system (Maples, Ash, etc.) will be a little easier once the ground around them has been dug up some. I read once where most of a tree's root system is in the first 2' underground and over 50% of a trees root structure is in the first 6". So, some digging will obviously help.
The hardest trees I have found to remove/knock down around my place are the $%^&*() "sweet gum" trees. They are very flexible with a strong tap root. They will usually just bend until they break off, no root ball to push over to get rid of the stump. Then you have to dig out the root which can be pretty deep. They are a pain.
Eddie