There are so many different species of tree and bird named in this poem that it's difficult to list them all. What are all these trees and birds doing in the poem? They're more than just pretty landscape, that's for sure.
For one thing, they could add to the important natural setting of the poem—like the farms and countryside, the trees and birds remind us of cycles of life: trees lose their leaves in the fall and they grow back in the spring. Birds lay eggs and have chicks in the spring. And in a poem about death and mortality, remembering that leaves do grow back and new baby birds are born every year is important. Not only might they represent the cycle of life, but specific types of trees and birds have different traditional symbolic meanings in Western poetry. Let's look at a few examples…
Line 10: Here's our first bird! It's an owl. The speaker personifies the owl when he says that it's "moping" and "complaining" to the moon. Since owls are nocturnal, they're often associated with death and with spooky hauntings. How appropriate for a poem about death that is set in a graveyard!
Line 13: Here are our first trees: elms and yews. Elms tend to be associated with strength in poetry (which may be why the speaker calls them "rugged"), while yew trees often represent eternity and immortality. It's not clear whether or not Gray intends to bring up the traditional poetic symbolism of these trees, but "eternity" sure would be appropriate, given that his poem is about death and what happens afterwards!
Lines 18-19: More birds! First he imagines a twittering, tweeting swallow, which is often associated with farms and barns, since that's where they like to build nests. Swallows are also early risers, like the "cock" or rooster that the speaker imagines crowing in the following line. These are the birds you hear first thing in the morning. The speaker is imagining the deaths of the local villagers, so these are the birds that he says they'll never wake up to hear again.
Line 101: Another tree—this time, the speaker is imagining how he'll be remembered after he dies. He thinks that folks might recall how he used to stretch out lazily under a beech tree. The beech is traditionally associated with ancient history, the written word, and knowledge of the past. Sounds like a great tree to associate with a poet, don't you think? What kind of tree or bird would you associate with yourself? Why?