That's an ambitious project!
Does it belong to you?
IMHO. the spaces around the tree trunks are 'way too small. Depending on what species of tree those are, and how much of the trunk is below the deck, you may need a 4-inch space +
all around it.
I'd suggest a round plywood panel—cut in half—to keep stuff from falling through—some joists may need "tweaking".
You'd be amazed at how far a strong storm can move that tree!
Otherwise, it will strain the tree at the wrong place—damage the trunk—and likely shift boards of the new deck.
I've had to make "serial adjustments" around the Colorado Blue Spruce that's been allowed to grow through our deck for the last 58 years. About four feet of the trunk is "below deck", and storms have driven the upper trunk against the roof.
