Add reverse tree view to existing app

I have a Laravel based app that displays a treeview. I would like it to also display an inverse tree view.

e.g. if I have a tree of
a
L b
L c
d
L c

I would like an inverse view alongside the tree that displays

c
L a
L d
b
L a

Now I want to be able to add a node to either the original or the inverse and it reflects the changes.

So if I were to add a node to the inverse tree it would update the original tree

Inverse:

b
L a
L New Node

Original:

New Node (instance 1)
L b

New Node (instance 2)
L a

If I now click on "New Node" it will be highlighted for all instances in both trees. This will be synced with the database. If I now edit New Node the changes will be reflected in all instances. All this will need to be synced with the database.