A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.

中文翻译：

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分散树的不变子集和 Bonato 和 Tardif 的树替代性质

如果一棵树不包含作为子树的完整二叉树的细分，则它是分散的。我们表明，每个分散的树都包含一个顶点、一条边或一组最多两个由 T 的每个嵌入保留的端点。这扩展了 Halin、Polat 和 Sabidussi 的结果。如果每棵树都嵌入另一个树，则称两棵树为等态树，然后我们证明，要么与散布树 T 等态的每棵树与 T 同构，要么与 T 同构的成对非同构树有无穷多个。 这证明了Bonato 和 Tardif 对分散树的树替代猜想，以及对局部有限分散树的 Tyomkyn 猜想。