# Well-founded trees of any order

Suppose $$T$$ is a well-founded tree on $$\mathbb{N}$$, that is, a set of finite sequences of $$\mathbb{N}$$ closed under taking initial segments. Well-founded means that there is no infinite sequence $$(x_n)$$ such that for all $$k$$, $$(x_1, x_2, \dots,x_k)\in T$$. Put $$T_0:=T$$ and for any succesor ordinal $$\alpha$$ define $$T_\alpha$$ to be the tree obtained by removing the maximal elements from $$T_{\alpha-1}$$. If $$\alpha$$ is a limit ordinal, $$T_\alpha:=\cap_{\gamma<\alpha} T_{\gamma}$$. The order $$o(T)$$ of the tree is defined as the smallest ordinal $$\delta$$ for which $$T_{\delta}=\emptyset$$.

Can one provide a reference, or a brif explanation if it is not too complicated, for the following facts, which I found mentioned, without any explanation, in a paper.

• For any tree $$T$$ on $$\mathbb{N}$$, $$o(T)<\omega_1$$.
• For any $$\alpha<\omega_1$$, there exists a tree $$T_\alpha$$ such that $$o(T_\alpha)=\alpha$$.

Since $$T$$ is countable (and wellfounded), at each successor step you erase some node. Having $$O(T)$$ where $$\geq\omega_1$$ would require erasing uncountably many nodes, an absurdity.
Finally, fix a wellorder $$R$$ of $$\mathbb{N}$$ in ordertype $$\alpha$$ and define $$T_\alpha$$ to be the tree of strictly decreasing $$R$$-chains.