A tree's n-path ideal has linear quotients exactly when the tree avoids P_n+P_n and the L_{n,k} graphs as induced subgraphs, for every n at least 4.
Bouchat and Tricia Muldoon Brown,Multi-graded betti numbers of path ideals of trees, Journal of Algebra and its Applications16(2017)
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Trees whose path ideals have linear quotients
A tree's n-path ideal has linear quotients exactly when the tree avoids P_n+P_n and the L_{n,k} graphs as induced subgraphs, for every n at least 4.