Why is there a custom to hang up flowers and growing green things on shavuot?
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You guys are mixing up two separate minhogim.
One minhog is for grasses/flowers, as a commemoration of the teaching that on Har Sinai at matan Torah there were grass and flowers (recall that it was in the desert).
A second minhog is to have tree branches. That is related to the teaching in the mishna that Shavuos is day of judgement for fruit of trees (peiros hailan). Some congregations do not do it since the GR"A (Vilna Gaon) held that it should not be observed.
Okay, I'll give the full answer:
Flowers are not mentioned in any of the original sources for these minhagim.
As Mordechai said there are more than one minhag being jumbled together in this question.
A. Minhag to spread herbs in the shul and homes as a reminder of the simha (happiness) of the giving of the Torah. It is a rememberance of this event because there were grasses and herbs growing by Har Sinai (even though it was in the desert, as Mordechai pointed out.) That is why it was necessary for the command to be given not to graze the sheep and cattle near the mountain.
B. Minhag to distribute herbs that have a scent for people to make a bracha on and smell. I have not yet found a reason specific to Shavuos.
C. Minhag to set up trees in the shuls and homes as a reminder that on Atzeres (Shavuos) we are judged regarding tree fruit. Based on how this minhag is worded, it is no wonder that the G"RA was against continuing it! Imagine a "Shavuos Tree" in all of our living rooms and in our Shuls! This is the only minhag of the three that it is said that the G"RA was against continuing.