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The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is possible to specialize a tree of height $\omega_1$ with a ccc forcing iff the tree has no cofinal branch. The forcing is just finite approximation. In the case where the tree is Aronszajn (thin and without cofinal branches), there are other varieties of forcings (in particular, it is possible to specialize such a tree with proper forcings that do not add reals).

Question: Are there known forcings used to specialize fat trees that do not look like the Baumgartner poset? For example, one that is proper that adds no reals.

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