For a tree T, let σT be the set of all increasing sequences in T of elements of T, ordered by initial segment (s<t of s is an initial segment of t). Why isn't σT embeddable in T ? In the sense there is no f from σT to T such that s<t --> f(s)<f(t) ?
I saw in a paper of Todorcevic that it is because such an embedding implies a one-to-one function between T and Ord, but I don't see where is the contradiction..
Assume that there is an increasing map $i:\sigma T\to T$. Now inductively define
$$X_\xi =\langle a_\eta\mid \eta<\xi\rangle, \quad a_\xi=i(X_\xi).$$
Then $\langle a_\xi\mid \xi\in\mathrm{Ord}\rangle$ is a strictly increasing sequence over $T$, a contradiction.