Subtrees with small branching number
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Given a tree $T$ , we can view a branch through $T$ as a subtree which is $1$-branching, i.e. each point on the branch has exactly one immediate successor on the branch. This point of view allows us to generalize the tree property by asking whether a tree can have tall subtrees with small branching number. In particular, we can ask whether an Aronszajn tree can have finitely branching subtrees of height $\aleph_1$. This leads to a class of trees which lies between those of Suslin and Aronszajn trees. In addition, such trees can be characterized as those having the Lindelöf property with respect to a reasonably nice topology.