I should have thought it is worth discussing this actual example, looks like a willow tree to me with a growth defect that clearly indicates that his is going to happen at some time.
Is it the council's responsibility? I have an old cherry at the front of my garden that gives a wonderful display of blossom early every year. One year I found a spray paint mark on a branch that overhangs the roadside pavement. Week or so later an official type letter, stating that there were symptoms to indicate that it might break and I must take appropriate action and within a time frame. It came with an addressed envelope and form to be returned when I had done the required work.
The tree could be part of the wood or it could be in the boundary of the field, but surely it is the responsibility of some property owner. What level of supervision has that property owner shown in allowing this to happen and was the notch cut to allow the gate to be used done on his behalf.
It is notable in Harry Cotterrels 'common sense policy' on access published by that organization that represents the large supposedly professional freeholders of land and like to think of themselves as landowners, that there is no recognition that they have any duty, where the law requires them to recognized public access.
I have a few examples, quickly available pre-photobucket debacle, but one comes to mind. Style and steps from roadside through to field, with fingerpost, but fingerpost neatly cut off level with newly cut hedge and wood chips all over the step. Add this observation to other signs of contempt for the idea that people should be walking in that area and it was clear that the hedge machine operator was expressing the landowner's opinion.