Symbolism, Imagery, Allegory
When Ty first meets Gemma when she's 10, he leaves an "abandoned and tattered" (37.34) robin's nest on her windowsill. In the present day, he tells her that when he found it, he took it to be "a sign that a person could do something different" (37.36)—like, for instance, kidnap a teenage girl and take her to a remote region of Australia. Of course, that's not the way Ty sees it. To him, he's going to the only place where he's ever really belonged, and he's taking Gemma with him.
So, it might seem like kind of a bizarre gift to leave for the girl you're stalking and will eventually abduct, but in a weird way, the nest symbolizes the connection that develops between Ty and Gemma. For one thing, the robin's nest is literally a broken home. As Ty tells Gemma, robins are "fierce birds […] They'll defend their home to the death" (37.27). And death seems to have occurred at the nest he brings her since it's clearly abandoned and destroyed. In the same way, Ty breaks Gemma's family's home by stealing her, leaving her family empty and completely changed.
That might be the negative side of things, but the dream Gemma has while reeling from the snake venom points to her starting to understand Ty as a person. She dreams of looking in her own bedroom window and seeing herself as a 10-year-old girl asleep in bed, then leaving the empty nest on her windowsill. "And then, I knew," she says. "I was you placing the bird's nest. But I was also me, looking out, too. I was us both" (84.16). While Ty brings damage into Gemma's life by kidnapping her, she nonetheless comes to relate to him and understand what led him to do what he did. And, in this way, they're sort of twisted birds of a feather.