AmandaMD

I just moved into a house that has very mature trees, not only mature but also neglected in the past 5 years or so. From my own google search I think this is a weeping cherry tree. I saw that many branches didn't have any blooms on it so knew it needed a good trim. Once I started cutting some dead branches I realized half of the tree is dead. I was scared to cut off every dead limb thinking it will look worse than it does. What should I do? I circled the dead limbs in red because it's hard to see in the picture. I don't know if I should cut it down to the bottom where the new growth is or just trim the dead limbs at the top that have no blooms or just leave it alone. Thanks for any help and suggestions.