Get the length between connected curves from a list or tree

Hi,
I am having a tree with suppose 15 curves names C1, c2, c3,....c15

Now, C1-c3-c5-c6-c7-c8
c1 - c2-c6-c7-c8
"-" denoted connected . Connected in the sense the end point of a curve is the starting point of another curve.

How can I determine the shortest length between two a number of sets of connected curves and check whether they are connected or not.

The curve or Line Like Curves are either in List or as tree..

Thanks in Advance,
Yajnvalkya Bandyopadhyay