A strategy for Isolator in the Toucher-Isolator game on trees

In the Toucher-Isolator game, introduced recently by Dowden, Kang, Mikala\v{c}ki and Stojakovi'{c}, Toucher and Isolator alternately claim an edge from a graph such that Toucher aims to touch as many vertices as possible, while Isolator aims to isolate as many vertices as possible, where Toucher plays first. Among trees with $n$ vertices, they showed that the star is the best choice for Isolator and they asked for the most suitable tree for Toucher. Later, R"{a}ty showed that the answer is the path with $n$ vertices. We give a simple alternative proof of this result. The method to determine where Isolator should play is by breaking down the gains and losses in each move of both players.