on the mend vs out of the woods

on the mend

prep
  • Healing or recovering, as from an injury or illness. 

  • My foot is on the mend but it still hurts to walk. 

  • Improving or undergoing restoration to a previous, more favorable condition. 

out of the woods

prep
  • Out of peril; likely to recover or prevail over trouble; finished with the worst or most threatening part of a problem or illness. 

  • Used other than figuratively or idiomatically: see out of, the, woods. 

How often have the words on the mend and out of the woods occurred in a corpus of books? (source: Google Ngram Viewer )