In New Moon, Carlisle and Esme give Bella plane tickets to go visit her mother. Bella is uncomfortable with all of the attention that she receives on her eighteenth birthday, because it is the first birthday that she is officially older than the age at which Edward was turned into a vampire. Though Bella is unhappy with the fuss over her birthday, she finds this gift incredibly touching, causing Edward to wish that she had opened it in front of his adopted parents.
It is also interesting to note that Bella does not use the plane tickets until the beginning of the next book. After her birthday party, Edward leaves because he feels like he would be robbing her of her human life if he stayed. Bella's birthday forces both Bella and Edward to reflect on the significance of aging as it affects their relationship.