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I got 3 willow trees like this that the trunks of 2 of them look dead. It started from the top and came down halfway. Branches are fine, get new buds once a while. I'm not sure if I could just leave it like this or it needs to be cut to prevent damaging the entire tree. It's been like this for last 2 months or so.

Also, one of the trees leaves turning dark with red spots. Not sure what's going on with that!

Pics attached.

Any idea how to save these? Thanks in advance.

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Location: NY
Outdoor on a balcony (north side, morning and afternoon sun then shade)
Pretty cold in last few nights (50°F-10°C) - day (65°F-18°C)

Cold? 10C is not cold, Are you watering enough? I would say it needs more sunshine.

Willow need loads and loads of water. Mine stands always in a tray filled with water, in summer days it drinks it empty twice a day ( 5liter of water).
And don't mind the temp, I never saw a tree with a coat or a jacket....

Thanks. Yes, I give them plenty of water with a tray full of water underneath.
I meant it's gotten colder. Dropped around 15-20 degrees in a couple of days.

My biggest worry is the dying part of the trunk which I'm not sure to leave or cut it.

I don't see anything wrong with that trunk really.