Tree height for a tree 
is the vertex height of its
root vertex, where the vertex
height of a vertex 
in a tree 
is the number of edges on the
longest downward path between 
and a tree leaf.
In some contexts, including definitions of treedepth, root-to-leaf height is instead counted by vertices rather than edges.
A function to return the height of a vertex 
in a tree 
may be implemented in a future version
of the Wolfram Language as TreeHeight[g,
v], and a function to return the height of a tree
as TreeHeight[g].
