Thanks for the response! The ideal photo you shared for the even shape and care is definitely what I would like to see. I'm concerned the tree we inherited isn't in the best of health, but I would like to see that change however possible.
The video you shared was excellent. My main concern is that the "left" side of the tree is clearly less dense and currently has several dead branches. That's readily evident now that it's flowering. Pruning that side further, although it needs it, will exasperate the lack of balance even further. This is why I wondered if the healthy side should also be trimmed back to help even it out the aesthetic.
@LibertyLizard@slrpnk.net indicated in another comment not trim back the healthier part unless stability was the concern. I wouldn't call it "unstable" based on the ground space and roots, but the trunk isn't entirely vertical and leans a bit towards the fuller side. Not sure that would classify as a "stability" issue though. It's not yet a dense tree.
I will check the soil this summer and water as needed as you suggest. Would you suggest anything else, perhaps certain kinds fertilizer, to help the tree?
When you said, "Leave it alone for now and then go for it in early spring." Do you mean next year? Or next month? The photos were taken the day of the post. It's very nearly "officially" spring and the tree is blooming.
Thank you so much for your response! Based on your light touch description, now is probably not the time to prune the dead branches from the more bare side. I agree that the tree doesn't seem unbalanced, albeit the trunk isn’t entirely vertical and leans a bit towards the fuller side. I agree that it's lack of density isn't likely the cause or an issue.
The bareness of one side is curious to me. I wish I could contact the previous owner. I also wonder if the magnolia tree, offset about 40ft/12m, has a part in that. It's about 45ft/13.75m tall itself. The magnolia is far off enough so as to not shade the weeping cherry at all, but perhaps the resource competition could be an issue?
What time of year do you recommend pruning of dead branches?
I don't suppose you know of any magic that will coax the sparse side to start produce more?