We've got a double whammy here: trees themselves are an important symbol of nature, but there's one special tree: the giant that Billy has to chop down on the first night he goes out hunting.
Most people see a giant, old tree, and they want to admire it, maybe take a goofy picture of themselves trying to get their arms around it. Not Billy. Billy wants to cut it down.
To be fair, the giant tree is Billy's test of character. It symbolizes (again) his dedication and persistence. Billy must conquer this tree to prove his loyalty to his dogs—not to mention, his own abilities to himself. And this is one giant tree. So, he almost gives up.
"Come on," I said to my dogs. "There's nothing I can do […] I can't climb it. Why it's sixty feet up to the first limb and it would take me a month to cut it down." (8.92-94)
But as we soon see, Billy will do anything for his dogs. So he grits his teeth, grabs his ax and takes on this test of faith, blisters and all.
Not only is this a physical test, but an emotional one as well. Billy must say goodbye to his life as he knew it before his dogs: "I knew I would miss the giant of the bottoms, for it had played a wonderful part in my life" (9.135). The tree represents a piece of his childhood that he must cut away in order to grow and become an adult. With each adventure and danger that he faces with his dogs, Billy moves farther away from childhood. The chopping down of the tree is his first step on that path to adulthood.
We just hope that the family at least uses it for firewood, or something.
But it's not all giants of the bottoms. The woods are full of trees, particularly sycamores: "All around me tall sycamores gleamed like white streamers in the moonlight" (8.43). More than any other tree in the wood, sycamores represent something divine with their whiteness, purity, and innocence. Whenever Billy gets scared he tends to hide behind the sycamore tree, rather that any other tree, and even uses them as shelter to rest and talk with his dogs—and with God:
I walked back to the sycamore tree. Once again I said a prayer, but this time the words were different. I didn't ask for a miracle. In every way a young boy could, I said "thanks." My second prayer wasn't said with just words. All of my heart and soul was in it. (11.78)
Need more proof? When Billy is searching for names for his pups. He finds the answer in…wait for it…a sycamore tree.
There carved in the white bark of a sycamore tree was a large heart. In the center of the heart were two names, "Dan" and "Ann." […] I stared unbelieving—for there were my names. They were perfect. (6.8)
Billy believes God has him these names the same way he answered his prayers for the pups. God's paper? The bark of the sycamore tree.