In Adrienne Rich's poem "The Trees," the addressee and subject of the long letters remain unmentioned. The short answer, therefore, is that they are written to whatever purpose and whatever person the reader decides.
Here is an attempt at a slightly longer answer that may do something to illuminate the poem. The lines in question are:
I sit inside, doors open to the veranda
writing long letters
in which I scarcely mention the departure
of the forest from the house.
The poet is sitting inside but facing outwards, with the doors open. In an echo of this stance, the long letters she writes are mainly, but not entirely, "internal." She scarcely mentions the departure of the forest from the house, but she does not omit it altogether. These, therefore, are not business or professional letters. She is unlikely to be regaling her bank manager or her accountant with elaborate metaphors concerning "long-cramped boughs shuffling under the roof." They are letters to people to whom she might mention such things, but only as much as they concern her personally: close friends and family. The fact that the letters may include hints of the trees, but only as background to the poet and her life, suggest that she is writing of her experiences, thoughts, and feelings in the sort of leisurely personal missives people used to write before we all got e-mail.
Rich's speaker, therefore, is preoccupied by her daily life and herself, as we all are. The trees are merely a background to her as she writes her letters. Being a poet, however, she does not leave matters as they are but returns to place the trees in the foreground as she writes the poem, placing within it a subtly ironic image of her preoccupied self, only faintly and peripherally conscious of what is now the principal subject of the poem.