As far as I know only a few plants have shown an ability to count or demonstrate the recall needed for basic pavlovian learning and none are trees- they are venus fly traps (counting), thale cress* (a type of division), and Mimosa pudica (recall and respond to stimuli appropriately).
The bit about mychorrizal fungi/”wood wide web” is true and it’s been noted to improve plant growth by “inoculating” the roots with a certain types of fungi for years- we’re just starting to figure out exactly why.
And while I wasn’t able to find a good source on the tree stump fact it is something that has happened, and I’ve seen happen, but I don’t know if I would say it is something trees “regularly” do. Or we’d see very few dead tree stumps.
Also, tree “individuality” is a little different than it is with animals and once two trees have grown together (via roots or any other connecting point) I don’t know if it makes so much sense to say they are healing (”keeping alive”) a fallen companion so much as they are healing what has effectively become a part of themselves. They started from two seeds but they are “one” tree now… so naturally they want to heal any weakness including a part of themselves that has “died”.
*thale cress can also “hear” and will become distressed if you play them sounds of caterpillars chewing on leaves.
also corn roots can “hear” the sound of other roots growing and will turn towards them to try and outcompete for space before the other roots get established