Aside from being fun to say, the Yggdrasill looks like a really cool place to live. Sophie and Josh realize as soon as they see Hekate's home that it is ancient. It's millennia older then the oldest tree on earth, the Methuselah tree in California. Inside, everything is made from natural materials—everything from the beds to the fur rugs, and nothing electronic works, because it would drain the tree's magic energy.
The Yggdrasill is immensely awesome; it's powerful, beautiful, and even seems to be its own ecosystem:
From just inside the entrance, [the twins] could look straight up to where wispy clouds gathered inside the tree […] dozens of tiny waterfalls sprouted from the walls and splashed down onto the floor far below, where the water gathered in a huge circular pool that took up most of the foyer. (15.11)
What's even cooler is that the Yggdrasill is also called "the World Tree." The combination of its old age and the mythology surrounding it indicate that it represents a state of wild magic and pure nature before humans came to muck it all up.
And most importantly of all, the tree is tied to life—Hekate's life in particular. When it falls, all the life within it turns to dust. In fact, the entire Shadowrealm world collapses without it. Talk about a reason to go green.