I am having difficulties with this problem.

Let T be a tree with k leaves and set [tex]t=ceiling(k/2)/MATH]. Prove that there exists paths which satisfy the following

i)

ii) for every i<j.

EDIT: going over my work i realized that my proof doesn't hold. So I'd appreciate any help on where to start. Induction seems to be the natural choice but i can't get it to work for any parameter.