Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif

A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.