While certainly many may argue that it is not appropriate for a Jew to have a Christmas tree, what is the exact prohibition of it? As far as I have heard, it is not an object of worship

ב"ה Shalom Your distinction that an x-mas tree is not an article of worship is correct, and because it is mainly a decoration and not worshipped, for certain intents and purposes its status as idol worship is lessened. (שו"ת שבט הלוי חלק ב סימן נז, שו"ת שבט הלוי חלק י סימן קלב, גם אני אודך- פסקי תשובות סי' י"ד, הרב גמליאל רבינוביץ, בשם הגרנ"א אהלבוים מח"ס מנחת חן, שו"ת אשר חנן חלק ו-ז סימן לא) However, having one or placing one in one's home or the like, touches upon another issue of not following in the ways of the gentiles, which is a Torah prohibition. The underlying thought behind this prohibition is to make a distinction between the Jewish people and the gentile nations, which is so understandable in our times when assimilation and intermarriage are so rampant. From that point of view, the Vilna Gaon even forbade decorating Shuls with trees and shrubs on Shavuot because of the similarity to the non- Jewish custom of decorating with trees. This opinion is mentioned also by the Chayye Adam, Mishna Berura and Aruch Hashulhcan. The Aruch Hashulcan even mentions that the custom of decorating Shuls on Shavuot with trees virtually disappeared from many communities due to the ban of the Vilna Gaon. (חיי אדם חלק ב-ג (הלכות שבת ומועדים) כלל קלא סעיף יג, משנה ברורה סימן תצד ס"ק י , ערוך השולחן סי' תצ"ד סעיף ו. ) Though, we see today that most communities are not stringent in regard to the use of trees and shrubs for the decorating of Shuls on Shavuot because there is a clear reason for this custom,(either to remind us of Mt. Sinai which was green with shrubbery or because we are judged on the fruits of the land on Shavuot) nonetheless the prohibition of following in the ways of the gentiles remains especially for an x-mas tree which is a clear symbol of the Christian holiday and all that it stands for. All the best