Rooting hormones work on basically every plant structure and it's been proven to increase elongation in existing roots over and over.
Would your tree benefit from it? A little. Probably.
Would it prevent death if the tree is on the verge? It could. But then again it also couldn't.
Auxins in their many forms can be a great aid - and a great nuisance in the sense that they elongate branches and increase foliage size. But the take home message is to not lean on them. They can't save what doesn't want to be saved, and they don't even work on some plant species at all.
Willow water contains more than auxins, it's filled with exudates, phenols, salicylic acid and other wound-healing chemicals that all combined do more good than auxin alone. But the snake in the grass is that willows can handle huge amounts of bacteria in standing water that could kill all other trees, especially the ones with wounds. So if you're going to use it, make sure it's fresh.