I can speak about Cougar.
After some Cougar south of town here had been treed many times, during a research project, they started to "fight back" a bit and would not "tree" so easily.
Before that, typically they were not aggressive.
The areas cougar are "run" here are completely forested so they have easy access to escape by climbing trees.
Other areas may be quite different.
The method sounds about the same.
Let the dogs out, listen to the baying, run like crazy, get to the dogs while they are at the tree (before they get clawed), decide and shoot or move on.
One addition here: you have to run through snow (typically)
The only people I have heard being "attacked" by cougars are small people and children being identified as prey in high population areas on Vancouver Island.
I have never heard a report of a hunter being attacked by a Cougar, followed by one sure, but never attacked.
Your biggest worry around here is for your lap dog (pet).
Leave him out on the porch in winter and a young Cougar is likely to turn it into lunch.