Retaining wall next to tree--how to fix or rebuild?
The question was bumped, so this has probably been overtaken by events.  But I'll add a couple of thoughts for anyone else in a similar situation.

My first thought is to question the tree root cause.  That's a pretty massive concrete wall.  If it had proper footers, I'm not sure roots could tilt it.  The growing roots would expand in the easiest direction, which would be into the soil.  I doubt that the roots could directly push the wall out of plumb.

Tree roots do create channels in the soil for water, and it's common for roots to result in the supporting soil being eroded.  If the roots are the cause of the tilting, that seems more likely.

If you really want to retain a concrete wall in that area (as opposed to say replacing it with a section of wooden fence to match what's visible in the second picture), there are a few ways to do it.  One is to re-support the wall with helical piers.  Another is to jack the wall back into position and get some good footers under it.  Since you don't want to damage the tree roots, that can require a creative solution based on exactly where the roots are, so it wouldn't be practical to speculate here.