Prophetic literature is full of poetic and figurative language that is not to be taken literally. The metaphor that Ezekiel is using here to describe Assyria (the greatness and then the fall) is that of the "Cosmic Tree."
In the ancient Near East, the cosmos was often envisioned as a tree. The tree was seen to be a center pillar, or axis mundi, of the universe.
From the Anchor Bible Dictionary (Vol. 6, pg. 658):
The description of the "world tree" or "cosmic tree" in Ezekiel 17:22-24; 31:2-9, and Daniel 4:7:9...should be noted. Such trees are portrayed as objects of beauty, grandeur, and abundant fertility. They become shelters fro all sorts of creatures and are a source of sustenance for all of life.
This picture of the cosmic tree is then used - as it is here in Ezekiel 31 - to describe a great kingdom (or the king himself). A kingdom so great that, just like the cosmic tree, it provided shelter and food for many under its protection. But, if the tree becomes arrogant, God will bring it down due to its pride.
In Daniel 4 - the Kingdom is Babylon. In Ezekiel 31 - the kingdom is Assyria but the metaphor is pointed at Egypt. In Ezekiel 17:22-24 - it is God who plants a kingdom in Israel.
The tree symbolizes the divine world order maintained by the king himself.
Since Ezekiel is using figurative language we are not to read that the trees of Eden were literally jealous. In describing the tree that represents Assyria, Ezekiel is using hyperbolic language that even the trees in God's own garden - Eden - were jealous.
The point though is that eventually God felled this tree. Since this metaphor is directed at Egypt - the warning is that even a kingdom as great as Assyria will come down and so will Egypt.