Beckett himself states: "No symbols where none intended" HOWEVER scholars, directors, readers, literay critics alike have all discarded that bit of advice and attempted tgo make sense of Beckett's nonsense (As a theatrical director and graduate literature major I am guilty of this myself with his Endgame). In my opinion, therefore, the tree is definitely a haracter in the play. In fact, it is the only character that experiences a positive growth and transformation. By act two, it has grown leaves - a sign of life in a world that is otherwise dead, deserted, abandoned and without sense. The tree has behaved as a tree should behave; whereas the people in the play do not behave as we would expect them to. The tree grows and changes, but not as a result of experience. It grows as a result of nature. Didi and Gogo, on the other hand, remain the same. They, too, are untouched by experience but they are also not transfoirmed. They continue to exist in the same state of limbo as they have from the start.
It should be noted that there is a great deal of difficulty in attempting to read into the tea leaves of the play. Its elusive nature is probably the intent of Beckett, himself. The tree can serve as the only character that functions in a clear teleological development over the course of the play. In the first act, it is barren. The characters debate whether or not it is a tree, shrub, or whatever. Even it is the subject of questioning and analysis in the play. In the second act, it has sprouted a few leaves. It is at this point that the assertion, "Maybe we should hang ourselves" is present. The tree symbolizes the passage of time, and how, like Heidegger says, "as soon as a man is born, he is ready to die." The only constant in the play is the passage of time, and the only "character" that clearly reflects the maturation and process of change is the tree.
The tree is an important part of the stage decor of Beckett's Waiting for Godot, all the more because there are so little stage-props in the play or for that matter in any other Beckett-play.
It is functional in many ways. It is a symbol of immobility and the static condition of human existence as exemplified in the illusory movements of circularity in Pozzo, Lucky and the two tramps who move only to reinforce their immobility.
The tree, like the minerals in Watteau's paintings, greatly admired by the young Beckett, mingles with the human condition manifested through the play.
The appearance of the two leaves in the leafless tree in the second act is a critique of temporal linearity and a subtle dig at a symbolic cliche of optimistic rhetoric.
The tree is also co-relative to the emptiness of the country road. It evokes a generic landscape, which can be so many places, almost any place in the world. And that is what de-spatializes the setting, in terms of specificity.