The ternary betweenness relation of a tree, B(x, y, z), indicates 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 greater 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 have called quasi-tree 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 be-tweenness 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. All trees and related structures are finite or countably infinite.