*Definition* A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T|y<x\}$ is well-ordered.


**Question** Assume CH fails and let $\aleph\_1\le\kappa< 2^{\aleph\_0}$. Is it possible that a tree of countable height and countable levels has **exactly** $\kappa$ many maximal branches through the tree? 



Tree with countable height and exactly $\aleph_1$-many maximal branches
No, and this is because every uncountable closed set of reals contains a perfect set. This is how Cantor proved that the continuum hypothesis holds for closed sets. 


The easiest way to prove it is by using the [Cantor-Bendixson analysis](https://en.wikipedia.org/wiki/Derived_set_(mathematics)#Cantor.E2.80.93Bendixson_rank): if you iteratively cast out isolated points from your closed set, then in countably many steps you'll arrive at a perfect set. So every closed set is the union of a countable set and a perfect set. Every perfect set has size continuum. 


