I've received a number of questions regarding Bilbo's material in the AUJ score, so perhaps a simply sketched guidebook is in order. Below is a listing of Bilbo's major thematic connections. These titles are just placeholders for the moment, so don't start canonizing them just yet!
The theme heard at 0:15 in the whistle in "The World is Ahead" relates to Bilbo's adventure. (A-B-C#-E-E-G#-G#-A-E-C#-E-D). This theme is first heard in "My Dear Frodo" at approximately 0:37 and comes right on the heels of a gesture in the strings and woodwinds that hints at the History of the Ring theme without ever fully articulating it (Bb-D-F over Bb-major; F-F# over B-minor, etc.) In my opinion, really sets the tone for the score to come. SPOILERS: You also hear this theme near the end of the film when Bilbo leaps in to protect Thorin from Azog.
This two-part theme is easily heard in "Dreaming of Bag End." The Baggins phrase appears at 0:10 and the Took phrase takes over at 0:33. It's kind of a single theme and kind of not, which is why I've been hedging my bets and referring to it as one-and-a-half themes. It relates to Bilbo's two primary impulses in the story: the urge to return home and the urge to go adventuring. As his character develops, the two halves begin to function independently. The Took phrase is often orchestrated for solo French horn, which is an unusually bold sound for Hobbit-based music.
The "Fussy" Bilbo theme is the triple-meter figure that alternates between D-minor and A-major. That major chord is what I'm on about in the Tracksounds podcast (which you can hear HERE, if you haven't already). It's a more classical sound that we're used to in Middle-earth ... more a part of historically functional harmonies, whereas Middle-earth usually traffics in modes or chromatic leaps. It's just a bit stuffy and put-upon sounding. This theme shows up about half-way through "Axe or Sword," but you also hear in the Trolls sequence.