
As you mention, Ajuga can indeed spread and creep into new areas quickly. To prevent this, you could add edging around the border in most locations, but I am concerned that this could damage the tree roots
These articles have some ideas but most of them will require regular upkeep:
https://www.gardeningknowhow.com/ornamental/foliage/ajuga/how-to-control-ajuga-plants.htm
https://www.gardeningknowhow.com/edible/herbs/mint/invasive-mint-how-to-kill-mint-plants.htm