That'll do it
Although, as you know, you can sometimes shoot one with more than one round of a .22 rim fire,
before they die.
Once they fall out of the tree, you know they are dead, not before
A funny story, from years ago. . .
We had both free range hens and horses on this land, which included woodland, and we were completely overrun with squirrels at one point because they had a limitless supply of animal food, which we kept in 17 IBC containers. We didn't mind feeding the squirrels, up to a point, but they used to chew through the containers, and each time they did that we lost both the container and the entire contents, ruined by the spillage.
So, my son walked around with his dog and a shotgun. Sometimes the dog got them, sometimes they got away from her and would run up a tree trunk, and he would then shoot them, and very rarely missed.
After a while there were very few left and it wasn't worth the time looking for them, so he set up a live trap. He visited the trap twice a day and whenever there was a squirrel inside he shot it, and then re-set the trap.
That was fine until his mother found the trap. She thought it was cruel, so she loaded the trap onto the back of the pickup whenever it had a squirrel in it, drove a couple of miles and released the squirrel. I told her that it was cruel to transport them, that it wouldn't work anyway because they would make their way back 'home' and that it was illegal anyway, but she knew best.
So, my son borrowed a few more traps and put them in places that his mother wouldn't find. And after a while, there were no more squirrels. His mum commented on the fact that there were no more squirrels and said "I told you that they wouldn't come back when I released them a couple of miles away". We didn't say anything and she never found out. She doesn't read TP
