Can I prune this...and can I prune it now? :-)

This tree (which Nobody on planet earth seems to definitely know what it is) is at least ten years old now. It is a grafted tree (is that right?) and is supposed to be pruned into a globe shape. It does flower very early on and the general consensus is that it may be some sort of cherry (no fruit though ever). It is now too high for me to reach to the top. I am about to get a new shed in a few weeks and am tearing down the old one tomorrow. Looking into the centre of this tree, the branches are very crowded. They are rubbing in places and I just wondered whether anyone can advise me on pruning. I'm not too bothered about the globe shape. Can I just cut away branches? I know this will then create gaps in the tree but am assuming these will just fill in, over time. Worse case scenario, if I was to cut it right back, would it sprout new growth? I've included two photos of it now and hopefully one which I took a while back when it was in leaf.

Thank you so much as always. The ones 'in leaf' are the best I've got....you can just about see it in each photo! The ones 'in leaf' were also taken over the last two or three years. It's grown quite a bit since then.
Mark

 

I would say it is a black cherry tree. Speaking on behalf of the cat, of course!

 

It's in the perfect spot I think...it looks lovely at the bottom of the garden. I just know that in six weeks time, when the new shed arrives, the fitters will say it's in the way for them to put up the new shed. I'm now thinking I may have cut it quite low but am hoping it will still sprout.
If not I'll have to cut it down to the stump. I'd be gutted :-( the pic is when I planted it. It was/is lovely ha ha

 

Hi Markymark, Lovely Black Cat. I agree with shiney, cut it back to accommodate the new shed, you may lose a few flowers/blossom but I'm sure it will pick up. Certainly take out some of the middle to allow the air to circulate.

 

Thank you so much. If I cut the developing buds off now which are right at the ends of the branches, I take it the will just form lower down? I will definitely have a go at it this week and hope for the best. I really want to avoid cutting it down. I'd miss it!!! Ha ha thanks once again.