  Depth first search is a natural algorithmic technique for constructing a
closed route that visits all vertices of a graph. The length of such route
equals, in an edge-weighted tree, twice the total weight of all edges of the
tree and this is asymptotically optimal over all exploration strategies. This
paper considers a variant of such search strategies where the length of each
route is bounded by a positive integer $B$ (e.g. due to limited energy
resources of the searcher). The objective is to cover all the edges of a tree
$T$ using the minimum number of routes, each starting and ending at the root
and each being of length at most $B$. To this end, we analyze the following
natural greedy tree traversal process that is based on decomposing a depth
first search traversal into a sequence of limited length routes. Given any
arbitrary depth first search traversal $R$ of the tree $T$, we cover $R$ with
routes $R_1,\ldots,R_l$, each of length at most $B$ such that: $R_i$ starts at
the root, reaches directly the farthest point of $R$ visited by $R_{i-1}$, then
$R_i$ continues along the path $R$ as far as possible, and finally $R_i$
returns to the root. We call the above algorithm \emph{piecemeal-DFS} and we
prove that it achieves the asymptotically minimal number of routes $l$,
regardless of the choice of $R$. Our analysis also shows that the total length
of the traversal (and thus the traversal time) of piecemeal-DFS is
asymptotically minimum over all energy-constrained exploration strategies. The
fact that $R$ can be chosen arbitrarily means that the exploration strategy can
be constructed in an online fashion when the input tree $T$ is not known in
advance. Surprisingly, our results show that depth first search is efficient
for energy constrained exploration of trees, even though it is known that the
same does not hold for energy constrained exploration of arbitrary graphs.
