I never had this fear of height in helicopters until after taking up parachute jumping for a hobby. Then it appeared sporadically and without any certain set of circumstances setting it off.
What I descibe it as being is not the fear of height but rather of being afraid of the fall that shall end with a real thump! It is not the fall but the thought of the sudden stop at the end.
While jumping with parachutes attached to your back....one's mindset is there will not be that sudden stop at the bottom as one of the two parachutes is bound to catch you short of that.
As long as you have confidance in your equipment.....then there is piece of mind.
Perhaps it is not a fear of heights but a crisis of confidance we are feeling.
But I do like the concept of the skinny branch and a subtle tie to monkey thinking as Robbo notes.