In　order　to　investigate　the　properties　of　weak　entwining　structures,　in　this　paper　we　firstly　use　the　methods　of　homological　algebra　to　discuss　the　weak　entwining　structure　of　a　path　algebra　and　a　path　co-algebra,　and　then　we　research　the　Hochschild　cohomology　of　the　weak　entwining　structure　and　its　properties,　and　obtain　equivalent　conditions　that　the　underlying　graph　of　the　path　coalgebra　in　such　a　weak　entwining　structure　is　a　tree.