The ternary betweenness relation of a tree, B(x,y,z) expresses that y is on the unique path between x and z. This notion can be extended to order-theoretic trees defined as partial orders such that the set of nodes larger than any node is linearly ordered. In such generalized trees, the unique "path" between two nodes can have infinitely many nodes. We generalize some results obtained in a previous article for the betweenness of join-trees. Join-trees are order-theoretic trees such that any two nodes have a least upper-bound. The motivation was to define conveniently the rank-width of a countable graph. We called quasi-tree the structure based on the betweenness relation of a join-tree. We proved that quasi-trees are axiomatized by a first-order sentence. Here, we obtain a monadic second-order axiomatization of betweenness in order-theoretic trees. We also define and compare several induced betweenness relations, i.e., restrictions to sets of nodes of the betweenness relations in generalized trees of different kinds. We prove that induced betweenness in quasi-trees is characterized by a first-order sentence. The proof uses order-theoretic trees.

翻译：B(x,y,z)树的永久间断关系表示,Y是在x和z之间的独特路径上。这个概念可以扩大到修饰性树木,被定义为部分的线性线性排列。在这种广泛的树中,两个节点之间独特的“路径”可以有无限多的节点。我们概括了上一个条款中取得的一些结果,以区分编织树之间的交错。联合树是秩序和理论的树,任何两个节点都有最小的上限。其动机是方便地界定可数图的分级线。我们根据连结树之间的交错关系而称之为准树结构。我们证明,准树是被一级句子分解的。在这里,我们得到了一个修配树之间分解的一阶。我们还定义和比较了两个节点之间的关系,即,即对可数图图的分界的分界系。我们称准树的结构是以连结的两端关系为基础。我们通过不同种类的典型的树木来证明,准树之间的分系的顺序。