  Marco Buratti's conjecture states that if $p$ is a prime and $L$ a multiset
containing $p-1$ non-zero elements from the integers modulo $p$, then there
exists a Hamiltonian path in the complete graph of order $p$ with edge lengths
in $L$. Say that a multiset satisfying the above conjecture is realizable. We
generalize the problem for trees, show that multisets can be realized as trees
with diameter at least one more than the number of distinct elements in the
multiset, and affirm the conjecture for multisets of the form $\{\phi_k(1)^a,
\phi_k(2)^b, \phi_k(3)^c\}$ where $\phi_k(i)=\min\{ki \pmod p, -ki \pmod p\}$.
