Geodesic
Automorphism Groups of Homogeneous Semilinear Orders: Normal Subgroups and Commutators
Canadian journal of mathematics, Tome 43 (1991) no. 4, pp. 721-737

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A partially ordered set (T, ≤) is called a tree if it is semilinearly ordered, i.e. any two elements have a common lower bound but no two incomparable elements have a common upper bound, and contains an infinite chain and at least two incomparable elements. Let k ∈ N. We say that a partially ordered set (T, ≤) is k-homogeneous, if each isomorphism between two k-element subsets of T extends to an automorphism of (T, ≤), and weakly k-transitive, if for any two k-element subchains of T there exists an automorphism of (T, ≤) taking one to the other. 
Droste, M.; Holland, W. C.; Macpherson, H. D. Automorphism Groups of Homogeneous Semilinear Orders: Normal Subgroups and Commutators. Canadian journal of mathematics, Tome 43 (1991) no. 4, pp. 721-737. doi: 10.4153/CJM-1991-041-7
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