In fact, for each infinite cardinal $\kappa$, there are $2^\kappa$ nonisomorphic trees of cardinality $\kappa$. The proof is by induction on $\kappa$.
Let $\kappa$ be an infinite cardinal. It follows from the inductive hypothesis if $\kappa\gt\aleph_0$, and is also true for $\kappa=\aleph_0$, that there are (at least) $\kappa$ nonisomorphic trees of cardinality less than $\kappa$. Choose a set $\mathcal T$ of $\kappa$ nonisomorphic trees, each of cardinality less than $\kappa$. Given any set $\mathcal S\subseteq\mathcal T$ with $|\mathcal S|=\kappa$ we construct a tree $T_\mathcal S$ of cardinality $\kappa$ by taking a new vertex $v$ and edges joining $v$ to one vertex of each tree in $\mathcal S$. To recover $\mathcal S$ from $T_\mathcal S$ we simple delete the unique vertex of degree $\kappa$.