Question 3 {4 marks}. Let T he a tree with 2!: leaves. Prove that there exist paths P1. P2, . . . . satisfying both properties below. 1. LJLlPl =T 2. V(H)V(-} ye llforevery 1 g 1' {j g in. Hint. As 2!: is even, there exists a collection of l: paths each between two leaves such that no leaf' appears in two different paths l[so far we do not assume these paths satisfy conditions {1] and (2}). Then use a maximality argument: consider a collection {P1, .. . . Pk} of such paths such that 2L1 |E[P;}| is mamnm among such collections and prove it satises properties {1} and {2}