If f1 is in F2 then take f2 = f1. EDIT as Casteels pointed out, this is ruled out by the wording of the question.
Else, T1 - f1 has two connected components say C1 and C2. T2 must have at least one edge from C1 to C2. Take that as f2.
If f1 is in F2 then take f2 = f1. EDIT as Casteels pointed out, this is ruled out by the wording of the question.
Else, T1 - f1 has two connected components say C1 and C2. T2 must have at least one edge from C1 to C2. Take that as f2.