That home recipe is going to burn the tops of the poison ivy and kill the leaves and green stems but won't kill the plants themselves. They will resprout in time. I doubt that any trees will be harmed by a light application of the product to the bark. If you get it on shoots they will be burned back.
Another inexpensive option is to use glyphosate (Roundup is one brand) as a cut stump treatment. You simply cut off the larger vines and then use a brush to "paint" the straight glyphosate onto the cut surface only. This doesn't take much product and works pretty well.
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