  Consider laying out a fixed-topology tree of N nodes into external memory
with block size B so as to minimize the worst-case number of block memory
transfers required to traverse a path from the root to a node of depth D. We
prove that the optimal number of memory transfers is $$ \cases{
  \displaystyle
  \Theta\left( {D \over \lg (1{+}B)} \right)
  & when $D = O(\lg N)$, \cr
  \displaystyle
  \Theta\left( {\lg N \over \lg \left(1{+}{B \lg N \over D}\right)} \right)
  & when $D = \Omega(\lg N)$ and $D = O(B \lg N)$, \cr
  \displaystyle
  \Theta\left( {D \over B} \right)
  & when $D = \Omega(B \lg N)$.
  } $$
