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.