The strong vertex (edge) span of a given graph $G$ is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of $G$ and moving either to an adjacent vertex or staying in the current position independently of each other. We introduce the notions of switching walks and triod size of a tree, which are used to determine the strong vertex and the strong edge span of an arbitrary tree. The obtained results are used in an algorithm that computes the strong vertex (edge) span of the input tree in linear time.