Does Christmas, which is supposed to be Christ birth and Christianity's most honored tradition, have pagan origins?

Should we celebrate a tradition that is based on a lie ( Christ's birthday is not Dec 25th...in the dead of winter) and is filled with worshiping pagan gods (the christmas tree is born)
Why it is that Christ never celebrated his own birthday although it had been a practive 5 centuries before Christ was born.
In all of the bible, birthday celebrations were only mentioned twice....(both were unconverted, non believers... and both times the storys were filled with violence and death. The first one mentioned
was the pagan worshiping Egyptian Pharoh who ordered his baker hanged....The second one was wicked king Harod who actually caused the death by beheading of John the Baptist as a birthday request. No, christ nor anyone following him engaged in this practice...so if we are christians and are supposted to "follow in the footsteps of Christ" should this be a holiday to observe?