# Well-founded trees on $\omega_1^{<\omega}$

As a motivation for this question, first consider the countable ideal $$I$$ of $$\omega_1$$, which has the following property.

(*) For every $$A \in I$$, there exists some $$A' \in I$$ such that $$A \subseteq A'$$ and for every $$C \in I$$ there is a permutation $$\pi$$ of $$\omega_1$$ such that $$\pi$$ point-wise fixes $$A$$ and $$\pi[C] \subseteq A'$$.

This is because for every $$A \in I$$ we can find a disjoint $$B \in I$$ and then $$A' = A \cup B$$ will do.

Now let $$T$$ be the tree on $$\omega_1^{<\omega}-\{\emptyset\}$$ (ordered by $$\subseteq$$) and $$I$$ be the ideal of $$T$$ generated by the well-founded trees whose first levels are countable. My questions are the following.

(1) Does $$I$$ have the analogous property of the countable ideal of $$\omega_1$$? Namely,

(*) For every well-founded tree $$t \in I$$, there exists some well-founded tree $$t' \in I$$ such that $$t \subseteq t'$$ and for every well-founded tree $$s \in I$$ there is an automorphism $$\pi$$ of $$T$$ such that $$\pi$$ point-wise fixes $$t$$ and $$\pi[s] \subseteq t'$$.

(2) Does any sub-ideal of $$I$$ have this property?