I always get a laugh at Marianne's pronouncement about Cowper's "beautiful lines which have frequently almost driven me wild" and her astonishment about Edward: "Nay, mama, if he is not to be animated by Cowper! "

I have a little book here of Everyman's Poetry: William Cowper. I'm afraid Marianne would find me quite as deficient as Edward, because I cannot picture being driven wild by any of his poetry.

Michael Bruce, in the intro. to this book writes: "one wonders if Jane Austen was not scoring a cheap shot at her heroine's tender, if rhapsodic, sensibilities" when she had Marianne make statements like that and like the passage you quote.

I haven't found a particular poem that this references directly, however addressing the trees in this manner is certainly something Cowper does, so it does sound like an imitation of him

This is from his poem 'Yardley Oak'

Time made thee what thou wast, king of the woods;
And time hath made thee what thou art—a cave
For owls to roost in. Once thy spreading boughs
O’erhung the champaign; and the numerous flocks
That grazed it stood beneath that ample cope
Uncrowded, yet safe shelter’d from the storm.
No flock frequents thee now. Thou hast outlived
Thy popularity, and art become
(Unless verse rescue thee awhile) a thing
Forgotten, as the foliage of thy youth.

...

Thine arms have left thee. Winds have rent them off
Long since, and rovers of the forest wild
With bow and shaft have burnt them. Some have left
A splinter’d stump bleach’d to a snowy white;
And some memorial none where once they grew.
Yet life still lingers in thee, and puts forth
Proof not contemptible of what she can,
Even where death predominates. The spring
Finds thee not less alive to her sweet force
Than yonder upstarts of the neighbouring wood,
So much thy juniors, who their birth received
Half a millennium since the date of thine.