Existence of infinite subsequence of trees with a subtree contained in the sequence

Question

Assume a statement:

For every infinite sequence of rooted trees $\{T\}_{i=0}^\infty$ there is an index $j\geq0$ such that there are infinitely many trees in $\{T\}_{i=0}^\infty$ which contains $T_j$ as subtree.

True or false? Could it possibly relate to The hydra game?

I need it for an extension of Dickson's lemma (and that I need for an extension of minimal coverability tree for membrane systems with active membranes).

Answer 1

What about the sequence of trees of which the first three are shown below? None is a subtree of any other.