The poetry of this scene introduces several important metaphors and symbols which will also recur throughout the play. When the Duchess uses the metaphor of a tree's roots and branches, to refer to the sons of the old king Edward III, she is using a very old metaphor which Elizabethans often invoked to describe their ancestral relations (and which we still use today when we talk about "family trees"). But the analogy between the royal family and the tree, with the dead Thomas being a branch "hack'd down, and his summer leaves all faded" (20), also introduces the idea that the royal family is linked to the natural world--and, specifically, that it is linked to the cycles of nature. Later, we will see other characters specifically refer to the way in that Richard's bad management of the country has left the crops dying and the plants withering.
We are also introduced to another of the play's central themes: the question of how a nobleman, or a king, ought to behave. When the Duchess tells Gaunt, "That which in mean men we intitle patience / Is pale cold cowardice in noble breasts" (33-34), she is bringing up the assumed differences between standards of behavior for commoners and the nobilitiy. The question of how a king ought to behave is a crucial issue for Richard throughout the rest of the play.