The plot of "Woodman, Spare That Tree!" can be summarized in the memories recounted by the speaker as justification for his refusal to allow the addressee to chop down the tree. The speaker says that the tree sheltered him in youth and that, long before this, one of his ancestors planted it near his home.
In the second stanza, the speaker claims that the tree is famous, known "o'er land and sea." He pleads with the speaker not to fell this massive, aged oak tree, which reaches up to the sky.
In the third stanza, the speaker reminisces about his childhood. He would seek out the shade of the tree. His sisters played here, his mother kissed him, and his father pressed his hand.
In the fourth and final stanza, the speaker addresses the tree directly, saying that his heartstrings cling to it, saying that he will never allow it to be harmed, and threatening the woodman, saying that he will do whatever is necessary to protect the tree from his ax.
The primary theme of the poem is nostalgia. The tree is valuable, not because of what it is but because of the place it has in the speaker's memory and the childhood memories it represents for him.