As soon as i read the title the first thought that came to mind is shedding old skin. This can be related to many things in general, trees (as you wrote) shed their leaves when they get cold and when the sunshines for spring they are renew. Snakes shed their skin as well. However
A keen eye will notice that this is a senryu and in that fact this poem is heavy in metaphors. As senryus deal with human emotions.
So in my mind the two key words in the poem are not something one might think. Trees represent a person in your life who is having a difficult time coping and slowly like the tree in winter (the tree is this person) they are slowly healing from a tragic event.
The wording surrounding the main plot points are well used. Because this is a poem of healing and you want to continue that emotion.