Why is it important to know that the balloon stayed stuck in the tree for a long time. (iReady)

If music be the food of love, play on; give me excess of it, that, surfeiting, the appetite may sicken, and so die. that strain again! it had a dying fall: o, it came o'er my ear like the sweet sound, that breathes upon a bank of violets, stealing and giving odour! now read the excerpt from "the love song of j. alfred prufrock." for i have known them all already, known them all: have known the evenings, mornings, afternoons, i have measured out my life with coffee spoons; i know the voices dying with a dying fall beneath the music from a farther room. what does the phrase "dying fall" most likely mean in both excerpts