Allegory
The theme of nature leads to great symbolism. The speaker encountering the bird has a relationship with nature, and when analyzed as allegory, it can suggest a deeper theme to this poem. Consider another poem by Dickinson called, "God Gave a Loaf to Every Bird." In it, you find the word "crumb" as something offered by God: "God gave a loaf to every bird,/ But just a crumb to me;/ I dare not eat it, though I starve,--." Just as in "A Bird Came Down the Walk," you can replace the speaker as God, and the speaker, as the bird. Suddenly, the theme of nature reveals another layer of the author's take on God.