A list of about 60 algorithmic problems pertaining to trees are given in the [paper][1] by Stephen T. Hedetniemi ([Wayback Machine](https://web.archive.org/web/20160308081109/http://www.akcejournal.org/contents/vol3no1/vol3_no1_1.pdf), [journal website](https://www.tandfonline.com/doi/abs/10.1080/09728600.2006.12088803)). Some of the discussion in the paper can be extended to questions on graphs of bounded [treewidth][2].  


  [1]: http://www.akcejournal.org/contents/vol3no1/vol3_no1_1.pdf "Stephen T. Hedetniemi & T.W. Haynes (Communicated by) (2006) Unsolved Algorithmic Problems on Trees, AKCE International Journal of Graphs and Combinatorics, 3:1, 1-37, "
  [2]: https://en.wikipedia.org/wiki/Treewidth