In The Night Circus by Erin Morgenstern, does Bailey merge into the tree?

No, Bailey does not merge into the tree at the end of The Night Circus.
Bailey is a great character.  It's through him that the reader learns to really feel the power and pull that the circus has on people.  Readers know that the circus is held together and made magical by the romantic overtures of Celia and Marco, but knowing that fact doesn't do the circus justice.  Readers need Bailey in order to feel and understand that the circus is as mystical and magical as we are told it is.  Bailey simply cannot resist the circus.  He is drawn to it.  He is drawn to sneak into it the very first time, and he is drawn to follow it across large distances with the other Reveurs.  He is so drawn to it that he willingly commits eternity to the circus. 
Bailey plays an important part in the story's ultimate climax.  The purpose of the circus is equivalent to a duel.  Marco and Celia are supposed to use the circus as their magical battle arena, and only one is supposed to come out alive; however, the two characters fall in love with each other.  In order to preserve their love and the circus, the circus's power must be merged/tied to a new individual.  Bailey is that individual.  Marco asks Bailey if Bailey is willing to be the circus's new caretaker, which is a permanent position.  It's permanent because everybody involved with the circus is immortal.  Bailey weighs his options, but he wants to be a part of the circus more than anything else in his life, so he agrees to be the person that the circus's magic is tied to and through.  Marco gives Bailey a ring and binds it to Bailey's finger.  Bailey's final step is to ignite the circus fire that has always stayed ignited.  He does this successfully, which guarantees the survival of the circus, Celia, and Marco.  Bailey continues to be the caretaker, in his human form, through the novel's end.