better vs out of the woods

better

verb
  • To surpass in excellence; to exceed; to excel. 

  • To give advantage to; to support; to advance the interest of. 

  • Had better. 

  • To improve. 

  • To become better; to improve. 

adj
  • Greater in amount or quantity 

noun
  • An entity, usually animate, deemed superior to another; one who has a claim to precedence; a superior. 

adv
  • Greater or lesser (whichever is seen as more advantageous), in reference to value, distance, time, etc. 

  • Had better. 

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 better and out of the woods occurred in a corpus of books? (source: Google Ngram Viewer )