Willow Woes

My neighbor has 2 beautiful willow trees that provide shade in my yard, and a source of visual enjoyment. Recently she has started cutting one of them down and plans to level both of them due to an enormous amount of tiny (2-5cm) red bugs, 6 legged, red bodied, with 2 little yellowish points on its back end. I mean tons of then, all over every branch. These little guys seem harmless, although messy as they spew a reddish color when flattened, and the tree is raining sticky sap as well. They dont fly, or bite. But the amount is overwhelming to her. I don't mind sweeping them off my deck, but I am so sad that these trees will no longer be a part of our backyard. Is there anything we can do? Can I tell her this will pass with the season, will the bugs return? or leave? Is her cutting down the trees the best solution? Thanks for your time, and patience in my less than knowledgeable state on these things.
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