Let $\kappa$ be some infinite cardinal. I usually use the notation $\langle T,<_T\rangle$ to refer to a tree. My question is: what is the number of non-isomorphic trees with $|T|=\kappa$?
There is an easy upper bound: $<_T\subseteq T\times T$, then, $<_T\in\mathcal{P}(T\times T)$, therefore, there are no more than $2^\kappa$-many non-isomorphic trees with cadinality $\kappa$. But, is this an optimal bound? Is it possible to define a tree for each element of $\mathcal{P}(T\times T)$? This question seems very basic and simple to me, however, I have not been able to find the answer anywhere. I suspect it may be independent of $\mathsf{ZFC}$, but I have no idea.
Thanks in advance!!
Given a set $S\subseteq\kappa$, consider the tree $T_S$ that has a maximal branch of length $\alpha$ iff $\alpha\in\kappa$. Concretely we can set $$T_S=\{f\in\kappa^{<\kappa}: f(0)\in S, length(f)<f(0), i>0\rightarrow f(i)=0\}.$$ Then clearly $T_S\cong T_{S'}$ iff $S=S'$.
(Fine, I technically need to restrict to $S\subseteq\kappa\setminus\{0\}$ and require $\vert S\vert=\kappa$, but there are again $2^\kappa$-many such subsets of $\kappa$ so that's benign.)