The line graph of a graph G is a graph having the edges of G as it's nodes and edges between them if the corresponding edges in G are adjacent. The Hamiltonian Completion Number is the minimum number of edges to be added to a graph for it to have a Hamiltonian Cycle.
Formally, the problem can be stated as asking for the Hamiltonian Completion Number of the line graph of a tree. While this problem is NP-Complete for the general case, it is in fact solvable in polynomial (linear actually) time for trees. Again, I do not have a simple algorithm, or a proof of why the algorithm works. Feel free to look at solutions or read up more about the problem online. See: http://en.wikipedia.org/wiki/Hamiltonian_completion http://www.sciencedirect.com/science/article/pii/S0020019000001642
Note that the caterpiller trees discussed above are precisely the trees for which the Hamiltonian Completion Number of their line graphs is 0.
Thanks for the code but please give some explanation about it.