is aware Christmas is not the birthday of Christ.

The Gospel accounts indicate that Jesus was born before the winter season:
Luke 2:8 "Now there were in the same country shepherds living out in the fields, keeping watch over their flock by night." {Sheep were never in the field by night in Palestine after the third week of October.} 

Inexplicable though it seems, the date of Christ's birth is not known. The gospels indicate neither the day, the month, nor the year. (The Catholic Encyclopedia, p.656, vol. 3, 1967.) 

What history tells us:
Despite the beliefs about Christ that the birth stories expressed, the church did not observe a festival for the celebration of the event until the 4th century. The date was chosen to counter the pagan festivities connected with the winter solstice; since 274, under the emperor Aurelian, Rome had celebrated the feast of the "Invincible Sun" [or Saturnalia] on December 25. (Christmas, The 1995 Grolier Multimedia Encyclopedia) 

Though the substitution of Christmas for the pagan festival cannot be proved with certainty, it remains the most plausible explanation for the dating of Christmas. (The Catholic Encyclopedia, p.656, vol. 3, 1967.) 

The decoration of the evergreen tree is of Pagan origin and predates Christ's birth:
Jer. 10:2-4 "Thus sayeth the Lord, learn not the way of the heathen ... for the customs of the people are vain: for one cuteth the tree out of the forest ... they deck it with silver and gold; they fasten it with nails and with hammers, that it move not." 

Man's customs and traditions, even if heartfelt, are not recognized by God as true worship:
Mark 7:6-7, Matthew 15:9 "Well did Isaiah prophesy of you hypocrites...in vain do they worship me, teaching as doctrines the traditions of men...thus making the word of God of none effect."