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You could install a wood deck over the concrete but you would need to build a frame first. The hardest part would be is that you would need to dig and install posts across the 28' span at least every 10' (with 2x8" joists), so your going to be digging up post hole size parts of the concrete.
You could cut the damaged concrete in the effected area, remove it and install a deck level with the remaining concrete, this would basically surround the problematic tree. Then you would have solved the tree root problem, you would only need to build a deck around the tree, it will look like you always planned it this way and you get to keep the concrete patio that's in good shape and have an artistic inlaid wood deck.
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