In pagan mythology, these two were twins of a whole. One couldn't exist without the other. Some think this folklore or belief was probably constructed by the Druids since they honored both trees; the Holly and the Oak.
The Holly King and the Oak King battles twice a year; once at Midsummer when the Holly King wins and then again at Midwinter or Yule, when the Oak King wins. They both fight for the favor of the Goddess. When one looses, he goes to Caer Arianrhod to lick his wounds for six months and then returns to battle again; exchanging places with the other. They are dual aspects of the Horned God.
Gods that is associated with the Holly King would be: Saturn, a Roman god of agriculture, Cronus, Father Ice or Grandfather Frost, Odin/Wotan, The Tomte, which is a Norse land spirit known for giving gifts to children at this time of the year & Thor.
Through time the Holly King became represented by the famous Christmas figure; Santa Claus or Father Christmas because he is depicted wearing red, a sprig of holly in his hair and is sometimes driving a team of eight stags, which could be a representative of the eight sabbats celebrated by the ancient pagans. He was also in some areas of Europe considered to be a powerful giant covered in Holly leaves and branches wielding a Holly bush as a club. In the Arthurian Legends, the Holly King could be seen as the Green Knight.
The Oak King is depicted as the Green Man all dresses in green appearing out from a foliage of green leaves. You can pretty much see him hanging over castle and church walls, starring down at you.
The Holly leaves are hung in honor of the Holly King and the mistletoe is hung in honor of the Oak King, because you can find mistletoe hanging far up in the branches of the Oak tree.
This time of the the year, the Holly King represents withdrawal, lessons, life and rest.
Take time this time of the year, while it's cold outside, go over the lessons of the year and see where to apply them to you life. Take the time to withdrawal and mediate on these lessons.
The two must battle for they are the cycle of the Sacred Wheel. The time of the Holly King is about over and the reign of the Oak King is upon us.