# Are the trees well-quasi-ordered by induced subgraphs?

It is well known that the finite rooted trees are well-quasi-ordered by the topological minor relation (a.k.a. homeomorphic embedding). Are they also wqo by the (induced) subgraph relation?

I'm confused because it seems to me that Nash-Williams proof can easily be adapted to induced subgraphs. This would be a stronger (and simpler) result.

Does anyone know a bad sequence?

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Consider the rooted trees shown below (with root at the top):

   *          *          *          *          *          *
|\         |\         |\         |\         |\         |\
* *        * *        * *        * *        * *        * *
|\         |          |          |          |          |
* *        *          *          *          *          *
|\         |          |          |          |
* *        *          *          *          *
|\         |          |          |   ...
* *        *          *          *
|\         |          |
* *        *          *
|\         |
* *        *
|\
* *


Clearly none of them is an induced subgraph of another, so this is an infinite antichain in the induced subgraph ordering of finite rooted trees, which is therefore not a well-quasi-order.

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