I read your comment first, and was expecting something far more complex. It's actually pretty straightforward, the basic idea is that the ephemeral Akyuu, whose life doesn't last, craves a connection to something that will endure, which is the tree she'd visited in previous lives. Keine, who also lives far beyond human years and has seen the village change considerably during that time, agrees to also be something that endures, that she can "come home to" when she reincarnates next.
The ink thing really doesn't seem to have much significance beyond a sign that she's going to die soon. It's a bit reminiscent of Tanyuu from Mushishi.
