There does not seem to be any particular reason why Henry Wadsworth Longfellow repeats the word tread specifically across these three lines which fall close to each other apart from the fact that he likes the sound of the word and the way it interplays with the surrounding sounds in his poem. This is a poem meant to be read aloud, and it makes great use of assonance and rhyme. So, we can look at each usage of the word in its particular context to identify why Longfellow may have selected it.
First, the "measured tread" of grenadiers is an example of assonance, with the e sound in both words creating a reflection of one another. There is also alliteration created here, with the word tramp in the previous line. "Tread" and "tramp" together create a very vivid image of the way the soldiers walk, suggesting a strong and military gait.
The next time the word tread appears it is also for reasons of soundplay. Here we again see assonance between this word and the preceding word stealthy. But we also see rhyme, with "tread" being rhymed with "overhead" in the next line.
Finally, the word tread is used as part of a simile describing the way the night-wind sounded, like a "sentinel's tread." Here, there is again assonance. We could argue that this is a very deliberate echo of the earlier description of the grenadiers' tread. It suggests that even the wind has begun to sound like soldiers due to the atmosphere and the tension of the situation.