Definition A tree means a set-theoretic tree, that is a poset (T,<) so that for each x∈T , the set {y∈T|y<x} is well-ordered.
Question:I would like to know if it is consistent with ZFC, the existence of a tree with height $\omega$ and each level of it has at most $\aleph_1$ elements, with maximal branches at least $\aleph_2$ but less than $2^{\aleph_0}$ (We assume the continuum hypothesis fails).
We know that if each level of the tree has at most countable elements the maximal branches are countable or $2^{\aleph_0}$, but I haven't seen any results if the levels are $\aleph_1$.

Definition A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
Question: I would like to know if it is consistent with ZFC, the existence of a tree with height $\omega$ and each level of it has at most $\aleph_1$ elements, with maximal branches at least $\aleph_2$ but less than $2^{\aleph_0}$ (We assume the continuum hypothesis fails).
We know that if each level of the tree has at most countable elements the maximal branches are countable or $2^{\aleph_0}$, but I haven't seen any results if the levels are $\aleph_1$.