While there are several reasons why there is a custom to spread greenery in shul, one which is brought by the Mishnah Berurah (494:10) is that Har Sinai was covered with greenery.
Why is it specifically this point about Har Sinai which we recall on Shavuos? Why doesn’t anyone, for instance, blow the shofar in shul, since there was a shofar at Har Sinai?