If the trees falling would endanger anything the girdling method of killing would be a bad idea.
I used it mostly on trees 6 inches and under that are winning the competition for sunlight selecting for what I want to win.
For instance I just girdled a few water oaks that are out growing some white oaks that I want for the final trees. The water oaks are about 5 inches in diameter or less and the whites just a couple of inches. The water oaks would win this race but for me and my hatchet.
I imagine the girdled trees sending out pheromones, etc, announcing their fatal wounding along with a decrease in allelochemical production. allopathic allelopathy allelochemical
A little human intervention over time goes a long way.
Another thing that I like about girdling is that is exposes the remaining plants to more light over a period of time Vs. instant full sun.