You can girdle them (strip away the bark about one inch wide and six inches above the ground level) and that will kill them. Then you have to cut and dig them out.

Something I've been wondering about for controlling trees/brush...

As an ecosystem changes over the long-haul, the ratio of fungi to bacteria changes as well. In a grassland, the ratio is typically given as 1:1. As you get into deciduous landscapes it goes up to 10:1. (Mature coniferous forests can have a fungal:bacterial ratio of 1000:1)

Is there an effective way through the use of teas or top-dressings to safely decrease the ratio so that the soil will favor grasses and forbs over trees and brush?

I don't know how helpful this is to your question, Jane99 - sorry.

Here is a link that might be useful: Soil Biology Primer - Typical Food Web Structures

Dr. Alex Shigo found that soils in forests tended to be more conducive to fungal growth because of the woody nature of the litter on the forest floor while the litter on a plains was more vegetative and that created conditions more conducive to bacterial growth which led to more grassy growth rather than trees.
This is not to say that tree seeds will not germinate and grow in grassy areas, they do in my lawn all the time, but mostly simply cutting those leaves off, which will then deprive the roots from necessary nutrients, will cause those roots to die.

The woody litter can cause a dominance in fungal biomass, and in turn, the woody plants specifically benefit from the fungi - more so than the grasses, right?

I'm not saying that you can't grow grass in a forest or trees in the grass, but isn't there a potential for tailoring the 'soil food web' to benefit one or the other?
Folks seem to have special bacterial and/or fungal-dominant compost tea recipes for different uses. Helping to reduce the hospitality of the soil towards a particular type of plant (i.e. trees) without harming the health of the soil seems reasonable.

It may not be practical to move the biomass ratio on a large-scale, and the actual benefit may not warrant the effort. But someday I'm going to look into it more - I have a lot of mesquite to practice on. Until then a chainsaw, pick-ax and chain will have to do.

(I wonder what soil conditions would favor grasses over prickly pear?)

I'd like to see that study done on soil under the care of some of the grass-fed cattle operations I know of. There is a lady north of Austin that runs 300 steers on 500 acres. She has (if I'm not mistaken) 30 pastures and each one has a different grass growing in it. She sprays 200 gallons of AACT every day and uses no other fertilizers or other chemicals. Her property would be perfect for a study like that.

Another operator in Mason, TX had 1,000 acres of prickly pear that he thought he could clear out by chain dragging. Needless to say he has the densest stand of pear anywhere. Over the years of leaving it alone, the duff has grown up around the pear and with the duff came the cochineal bug (coach uh NEEL is the English (mis) pronunciation). Cochineal is a parasite that destroys prickly pear. It is a little white puff that looks like cotton until you pick it off. Then it looks like blood. Cochineal is raised as a crop in Mexico to be used as a dye. This rancher's observation was that the cochineal did not come until the duff layer was pretty well developed. Apparently the bug has a development stage that requires protection from the elements. In any case he foresees the day when he'll be able to eliminate the prickly pear and put that acreage into production.