(5)→(6): is connected and if{x,y}∈E() then T-x, y is disconnected. We prove There is a unique simple path between any two of vertices of T” 4(6)(1): There is a unique simple path between any two of vertices of T. We prove T is a connected graph with no simple circuit &T is a connected graph 4 If T contains a simple circuit, then ☆ contradiction❖ (5)→(6): T is connected and if {x,y}E(T) then T-{x,y} is disconnected. We prove “There is a unique simple path between any two of vertices of T”. ❖ (6)→(1): There is a unique simple path between any two of vertices of T. We prove “T is a connected graph with no simple circuit” ❖ T is a connected graph . ❖ If T contains a simple circuit, then … ❖ contradiction