A search game with traveling cost on a tree

There is a rooted tree. A person selects a vertex except the root, hides in it and stays there. The searcher is at the root and then examines each vertex until they find the hider, traveling along edges of the tree. Associated with the examination are a traveling cost dependent on the distance from the last vertex examined and a fixed examination cost. The searcher wishes to minimize the expected cost of finding the hider, whereas the hider wishes to maximize it. The problem is formulated as a two-person zero-sum game and it is solved.