You must use Lifeweaver’s ultimate to heal 1,200 health points in one use. As of release, his ultimate is comparatively slow in healing, pulsing as it heals. An aura indicates its range. The tricky part will be getting your other support (assuming you’re attempting this in role queue) to not heal up your team, thus removing potential for your tree to do the healing. Anna’s biotic grenade and Junker Queen’s rampage will essentially shut this down as well. The tree can be destroyed and in its current state might actually be more barrier than healing utility. I got the achievement by healing my team up with the tree just before the next fight and having them pick at the other team while coming back to the aura. My video:
Lifeweaver Cute & Pixel Spray Guide: Giving Tree & Life Weaving Achievement
This should be unlockable in open queue. However these achievements/sprays are frequently glitched on release and only unlock in role queue. That said, worth giving a try if you are confident in your ability to select the new hero first. Multiple tanks will allow for a greater health pool to recover.