I had seen this past Shavuos in many shuls that not only flowers were hung around the shul but actual branches from trees that were seemingly cut down to put up in shul. What is the heter to cut branches from a tree during Shmitta in order to decorate a shul for Shavuos?
Cutting trees during Shmitta would be permissible as long as it is being done for the wood being cut down, whether for fire wood, decorative use or any other purpose. This is similar to picking fruit which is permissible during Shmita. The prohibition for cutting trees and branches is only when it is being done for pruning to improve the growth of the tree. Amateur cutting generally only damages growth and is clear to all that it is being done for the wood / tree which is permitted.