I'm pretty sure that table is using the Nested Sets design, not Adjacency List.  If it were using Adjacency List, it would have a column like parent_id instead of left and right.
Moving nodes is royal PITA in Nested Sets.  You have to renumber all the left and right values for each node you move.
If you move a subtree, the easiest way to do this is to remove the nodes one at a time, renumbering the left and right fields after each node removal.  Then once you have removed the whole subtree (and preserved the structure of the subtree in your application somehow), re-insert the subtree at its destination location in the tree, again re-numbering the left and right fields per insert.