  The contraction cost of a tensor network depends on the contraction order.
However, the optimal contraction ordering problem is known to be NP-hard. We
show that the linear contraction ordering problem for tree tensor networks
admits a polynomial-time algorithm, by drawing connections to database join
ordering. The result relies on the adjacent sequence interchange property of
the contraction cost, which enables a global decision of the contraction order
based on local comparisons. Based on that, we specify a modified version of the
IKKBZ database join ordering algorithm to find the optimal tree tensor network
linear contraction order. Finally, we extend our algorithm as a heuristic to
general contraction orders and arbitrary tensor network topologies.
