  We make progress on a number of open problems concerning the area requirement
for drawing trees on a grid. We prove that
  1. every tree of size $n$ (with arbitrarily large degree) has a straight-line
drawing with area $n2^{O(\sqrt{\log\log n\log\log\log n})}$, improving the
longstanding $O(n\log n)$ bound;
  2. every tree of size $n$ (with arbitrarily large degree) has a straight-line
upward drawing with area $n\sqrt{\log n}(\log\log n)^{O(1)}$, improving the
longstanding $O(n\log n)$ bound;
  3. every binary tree of size $n$ has a straight-line orthogonal drawing with
area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by Shin,
Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);
  4. every binary tree of size $n$ has a straight-line order-preserving drawing
with area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by
Garg and Rusu (2003);
  5. every binary tree of size $n$ has a straight-line orthogonal
order-preserving drawing with area $n2^{O(\sqrt{\log n})}$, improving the
$O(n^{3/2})$ previous bound by Frati (2007).
