Rowling revealed she connected the wands of Harry, Ron, and Hermione in a subtle way: they are all Celtic rooted.
Harry's wand is made of Holly, religiously and scientifically proven to be instrumental in guarding against lightning strikes. Holly is also the "tree of sacrifice" within folklore. We know the wand chooses the wizard and Harry had a lightning-bolt scar from the twin wand, made of Yew. Yew properties contrast Holly. The common cores of a Phoenix feather came from Fawkes. This seems to connect Dumbledore, creating a trio.
Ron Weasley had two wands. The first was handed down from his brother Charlie Weasley and was made of Ash with a unicorn hair core.
Ron's second wand is made of Willow, retaining a unicorn hair core. Willow maintains Rowling's theme of Celtic traditional woods amongst the series trio.
Hermione's wand is made from vine wood, with a dragon heartstring core. As with the wands of her friends, Rowling picked these based on Celtic correlation with their birthdays. Vine wood properties are a commanding authority, joy, and sacred knowledge.