Symbolism, Imagery, Allegory
It ain't Christmas without a Christmas tree, right? From the iconic one in Rockefeller Center to every drugstore playing some infuriating version of "O Tannenbaum," it seems like nobody can get through the (arguably) most magical time of the year without getting a good whiff of pine.
The Christmas tree itself is symbolic: it brings to mind all that is Xmas. Beyond that, however, it can be seen as being directly symbolic—in this play—of Nora. How, you ask?
First of all, the tree seems to mimic Nora's psychological state. At the beginning of Act Two, stage directions tell us, "The Christmas Tree is […] stripped of its ornaments and with burnt-down candle-ends on its disheveled branches" (2.1). But what does that have to do with Nora? Stage directions go on to say that, "[Nora] is alone in the room, walking about uneasily" (2.1). Basically, Nora is a mess and so is the tree. She's gotten the bad news from Krogstad, and as a result her mind is just as disheveled as the poor tree.
You could also interpret the tree's state as symbolic of Nora's disintegrating web of lies. The pretty decorations that Nora used to cover up her deceit are falling away. Soon the bare, ugly truth will emerge.
Lastly, Nora's function in the household is pretty much the same as the tree. She's merely decorative, ornamental. She dresses up the tree just as Torvald dresses up her for the Stenborgs' party. It's interesting that she tells the maid not to let the children see the tree until it's decorated. This is reminiscent of when she tells Torvald that she can't be seen in her costume until the party.
It seems that Ibsen built in many parallels between Nora and her tree—and none of them have to do with being crowned with a star and surrounded by presents.