Track intersected binary lines

Dears,

I would like to kindly ask you for help with tracking lines in binary image. I have two intersected lines in the binary image (see below), and I need to track individual lines. My idea is perform the image labeling, but I do not know how to do it when lines are not isolated. I will appreciate any help.

[darova]

What about bwmoroph? 'skel' -> 'endpoints' -> 'branchpoints'

[Jan Kubicek]

Thank you very much for the reaction. Can you please provide code for that, I am little bit confused how to do it. Many thanks.

[darova]

I want you do this