Q. Since the branches placed on Shavuos are from trees, is it permitted to place them in shul or even at home?

A. Maharil (H. Shavuos p. 160) is one of the earliest sources for placing aromatic plants and flowers on the floors of shuln and homes; he does not mention branches. Mogen Avrohom (494) does mention the placing of trees as a remembrance that in Shavuos we are judged on the attainment and success of the fruits of trees (Rosh Hashana 15a). Imrei Pinchos (p.148) mentions that following Cabala teachings, Mogen Avrohom means specifically trees and not branches.
However, the HaGro (Ma’ase Rav 191, quoted also in Chaye Adam 131: 13, Mishna Berurah 494,10)), rescinded the minhag of placing trees, since in our days it has become a Gentile’s custom. It is included in the ban of following Chukos Hagoyim, or a tradition of other religions. Another reason for not placing trees in a shul originates from the Biblical injunction against planting trees next to the altar in the Bais Hamikdosh. (see R. A. Eiger notes on O.C.150). Orchois Rabbenu (p.134) mentions that both the Chazon Ish zt’l and the Steipler Gaon zt”l did not place any greenery in their homes on Shavuos.

Many Other Poskim did maintain the tradition of at least placing branches, (Imrei Eish Y.D. 58, Ksav Sofer Y.D. 178, Maharsham 1: 127 et. al.) if not trees in homes and shuln. (Yosef Daas Y.D. 348 who permits trees also). Otzar Hachaim (Vayikra 186) opines that unlike other religions, we only place the trees inside the homes, therefore it does not resemble the traditions of the Gentiles.
However, Igrois Moishe (Y.D. 4,11), quotes Aruch Hashulchan (O.C. ibid. 6), that not placing trees has become today the accepted tradition. (See question 89 in this forum).
Horav Shlomo Miller’s Shlit”a opinion is that one should avoid decorating with complete trees. Nonetheless, placing plants, flowers and branches will depend on the minhag and tradition of the family or location.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a