On a tree, a single cop can always catch the robber. The cop uses a strategy called "vertex pursuit" that guarantees capture in moves. The cop moves toward the robber along the unique path between them. The tree structure prevents the robber from going around.
This is a classic result in graph theory. The proof involves analyzing the distance between cop and robber at each step. The distance strictly decreases every turn because the tree has no alternative routes for the robber.