  We study the algorithmic problem of optimally covering a tree with $k$ mobile
robots. The tree is known to all robots, and our goal is to assign a walk to
each robot in such a way that the union of these walks covers the whole tree.
We assume that the edges have the same length, and that traveling along an edge
takes a unit of time. Two objective functions are considered: the cover time
and the cover length. The cover time is the maximum time a robot needs to
finish its assigned walk and the cover length is the sum of the lengths of all
the walks. We also consider a variant in which the robots must rendezvous
periodically at the same vertex in at most a certain number of moves. We show
that the problem is different for the two cost functions. For the cover time
minimization problem, we prove that the problem is NP-hard when $k$ is part of
the input, regardless of whether periodic rendezvous are required or not. For
the cover length minimization problem, we show that it can be solved in
polynomial time when periodic rendezvous are not required, and it is NP-hard
otherwise.
