My understanding, and I hope I can explain this properly is -
Jesus was crucified.
We tend to think of ‘hanged’ as in by a noose round the neck - from a tree, and crucified as he is depicted on crucifixes - nailed through hands and feet - to a tree.
There is another understanding of those words e.g. hang a picture on a wall - usually the wire on the back of the frame is what ‘hangs’ on the nail or hook. So it can also be used in a similar way that Jesus was hanged to a tree ie put upon the face of the tree.
My other thoughts are that even though in movies etc we see and are familiar to seeing the cross as two rectangles of wood - sawn and shaped neatly etc. Back then the tools they had I don’t think would have been capable of the same result. Perhaps a rough shapened tree for the vertical ‘branch’ and the horizontal ‘branch’ was a smaller tree or branch. The wood of the tree which formed the cross upon which He was crucified and which we venerate each Good Friday.
It doesn’t mean Our Lord was nailed to a living tree.
I’m also thinking here of the Apostle Peter who was crucified upside down, and I think the Apostle Andrew was crucified on an X - so a living tree would not be possible, so trunks of trees etc would fit here.
I think these were all trees or logs or trunks of trees cut down upon which to crucify people on. Tree is what they were on, crucified is what was done to them, the form of punishment dealt out.
Not sure if I’ve helped or only muddied the water for you. Hopefully someone else will be along soon who can better explain it to you.