What is the significance of the tree in "Waiting for Godot"?
With Beckett, it's best not to get tied down to one particular meaning. At certain points, all that you can do is put forward your own interpretations and hope for the best. Having said that, some interpretations have more credibility than others. My own interpretation is intended to point you in a particular direction, but no more. So, what follows is not intended to be in any way definitive.
The tree is a very useful plot device in that it anchors Vladimir and Estragon to a specific place, a place where they are waiting for Godot. We learn that Godot told the two men to wait for him by the willow tree, so here they are. The tree doesn't simply provide a decorous background or a stage prop; it's an integral part of the play.
As mentioned by a previous contributor, the willow tree has great religious significance. In the Judeo-Christian tradition it represents, among other things, the promise of new life as well as chastity. But the willow symbol is even older than that, going way back to the strange, mystical world of Celtic antiquity. The willow is a symbol of harmony, of strength, of rootedness to the soil.
But Estragon's none too impressed with the willow. In fact, he questions whether it's even a willow at all:
Estragon: Looks to me more like a bush.
Vladimir: A shrub.
Estragon: A bush.
Vladimir: A–. What are you insinuating? That we’ve come to the wrong place?
Far from being a symbol of strength and stability, the willow tree has become for Estragon, at least, a source of uncertainty, an object whose integrity must be called into question. Vladimir soon joins Estragon in his questioning of the tree's positive symbolic value. No more is it a site of salvation; it's simply an object from which you can hang yourself. In this expression of existential ennui, it's not just the creation that's being called into question, but the Creator himself.
In the midst of this existential crisis, the integrity of memory begins to fracture. In act 2, Estragon is adamant that the tree has changed even though it seems exactly the same as it was the previous day. In his act of forgetfulness Estragon appears to be projecting his own fractured sense of self onto the tree; the unchanging stability of the tree stands not just as a contrast, but almost as an insult, to his own lack of selfhood. He cannot find a place for himself in the world, and so deeply resents any person or thing that has a firm connection to the soil.
By contrast, Vladimir initially sees the willow in more conventional terms, excited as he is by the new leaves that seem to prefigure hope. But ultimately, he succumbs to despair through all that endless waiting. By the close of the play, the tree is no longer a symbol; it is a thing, something that just happens to be there. No more, no less.
A traditional symbol is never just a mere symbol; it always points to something of much greater significance than itself. But Beckett has stripped the tree of whatever symbolic value it may possess, thus depriving it of any objective meaning. The way is now left open for the audience to project its own meaning onto the tree. And the same applies to anyone writing an essay about the significance of the willow tree in Waiting For Godot.
'A country road. A tree' is all the setting for WFG. And, in Act 1, the tree has no leaves:
Where are the leaves?
It must be dead.
No more weeping.
Or perhaps it's not the season.
Looks to me more like a bush.
You notice how, like everything else in the play, it changes and is hard to define and pin down. They're not even sure the tree is a tree. And in the second half, it sprouts "four or five leaves":
(Silence. He looks at the tree.) Everything's dead but the tree.
(looking at the tree). What is it?
It's the tree.
Yes, but what kind?
I don't know. A willow.
Estragon draws Vladimir towards the tree. They stand motionless before it. Silence.
Why don't we hang ourselves?
It's not a symbol that works in a lateral way. It might represent the gallows tree on which Christ was crucified (i.e. the "one of the thieves was saved" interchange), and it certainly is associated with hanging. Death might be an escape from waiting, or a redemption. A defeat, or a hopeful prospect. It just isn't clear.
And what are we to make of its sprouting leaves? Is it coming back to life? Is to simply to signify a lot of time has passed? Or is it a sign that hope has bled into the play - that it isn't entirely pointless? Make of it what you will. But it's all of these things, I think - a symbol of nature and of life.