Some excellent advice from the experienced hands here.
Use of the winch, or any other machinery on site, for restraint/crown rotation, work from the butt upwards, 'feel' how the tree reacts and see whats balancing on what, keep on the uphill side on a slope, etc...
I am, however, surprised that if you are new to this kind of operation your boss or an experienced colleague wasn't with you to advise you and show you how to go on.
Nevertheless, the only way you will learn how to judge these things is by having a go and soaking up the experience.
Also, interesting how different work managers assess risk and plan a course of action.
John Hancock said he would have been happier partially dismantling the tree in his pictures and felling a relatively striaght clear stem, to avoid men working under/adjacent to tensioned timber.
I would be happier to fell in one go and breakdown the crown in a controlled manner to minimise working at height.
Neither of us is right or wrong, just a different way of achieving a safe end result.
One thing that no one else has mentioned yet is planning direction of fell.
Not always possible, there may be no choice available, but sometimes you can get the crown to land in a better position for breakdown.
Pictures never tell the full story, but I would have thought that with the space available in John's picture, it may have been possible to fell that edge tree slightly left handed of where it lays.
This would have meant the crown would have landed right hand side first and the whole tree would have rotated a 1/4 turn anti clockwise as it came to rest.
The end result being the tree is laid trunk down, pretty much on the floor and the crown is laid to the right hand side and is a far easier and safer prospect to cut up.
Alternatively, space and equipment permitting, it could have been possible to fell the tree backwards with a pull and then the trunk would be on the floor with the crown uppermost.
Again, an easier and safer breakdown, than the one shown.