Can a cherry tree heal?

We have this sick cherry tree on our front lawn. It blooms beautifully in spring, but soon after blooming large holes appear in the leaves, some of the leaves get eaten completely, others crumple up. We found tiny black insects on the underside of the leaves, probably aphids.
Earlier this summer a tree surgeon from the area was driving past while I was fiddling about in the front garden and he stopped and asked if I wanted the tree removed by any chance - not only is it obviously sick but also it is getting fairly large (although I do like the size of it, nice bit of privacy at least when it is in leaf). I said not right now, maybe next year.

I don't know whether the tree overheard that bit of conversation, but almost immediately the appearance of new holes in leaves stopped. We had just two cherries on the entire tree last year, and they were tiny and distorted. This year, though still rather small, we have a lot of cherries in the tree, and the blackbirds love them (which, in turn, is nice entertainment for the cats to watch!). The tree, while still displaying problems, seems so much healthier than last year and I am thinking whether it is really necessary to chop it down or maybe, if this year is an improvement to last year, over time we might end up with a healthy cherry tree again?
Last year ALL the leaves were affected, no exception. Like I said, this year suddenly the process seemed to have stopped and the new growth at the top of the tree is completely unaffected.

What do tree specialists here think of that?

 

The tiny black insects sound like Cherry aphids. They may well have beebn reduced in numbers by a harsher winter or something. The migrate from the cherry to either Galium or Brassicas and then return so if something broke the cycle then you would see fewer of them on the tree. Obviously any improvement in the quality and quantity of the leaf would improve the health of the tree.