Alliteration is an important part of "The Black Walnut Tree," lending intensity to the speaker's words. As an example, take the alliteration in lines 6-7:
Likely some storm anyway
will churn down its dark boughs,
This quote comes as the speaker and her mother build a case for why they should sell the walnut tree. Not only would it pay off their mortgage, but it would also eliminate the risk that the tree poses to the house: a storm could easily send its limbs crashing through the house. Anticipating the violence of this yet-to-arrive disaster, alliterating (and sibilant) /s/ sounds convey the wind and the rain, while the two hard /d/s have a heavy quality suggestive of the weight and might of the tree.
Soon after this moment, the speaker notes how she and her mother:
[...] talk
slowly, two women trying
in a difficult time to be wise.
The chattering /t/ sound reflects all this talk. The /t/ sound is also found in the word "tree" itself, of course, perhaps works as a kind of subconscious echo, nagging at the reader.
Another important moment of alliteration appears in lines 16 and 17 as poem makes the turn from the argument in favor of cutting down the walnut tree to the more convincing one against doing so:
But something brighter than money
moves in our blood [...]
This alliteration sounds bright and loud, evoking the way that, despite the practical reasons for cutting down the tree, something deeper calls the two women in the opposite direction. The strong /b/ sounds here literally surround the /m/ alliteration of "money" and "moves," subtly reflecting how this "something" (a reference to respect for the past and the speaker's heritage) is more powerful than financial concerns.
Later, strong alliteration again appears when the speaker recounts a dream she has about preceding generations working the fields of Ohio: "fathers," "filling," fields," and "fresh" all share the initial /f/ sound, creating a sense of natural abundance and beauty that reflects the "generous" land; it sounds as if the lines themselves are overgrown with beautiful sounds.
Finally, the poem returns to the sibilant /s/ sound in its final few lines:
So the black walnut tree
swings through another year
of sun [...]
This reminds the reader that the storm mentioned in line 6 remains a threat, even though the two women have decided to keep the tree. Then, to make clear that the financial danger hasn't gone away either, as "month" and "mortgage" alliterate heavily in the last two lines.