Upper bound on the number of edges relative to the height of a DFS tree
After running a DFS, all the edges in $G$ can be classified as tree edges or back edges. Each back edge connects a vertex on the tree to one of it's ancestors. For each vertex it can point to at most $h-1$ ancestors, thus you can have at most $(h-1)|V|$ back edges. There are $|V|-1$ tree edges. You can have at most $h|V|-1$ edges.