Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity automorphisms of T. We give bounds on the maximum valence d of T that assure the existence of a vertex coloring of T with two colors that is preserved only by the identity automorphism. For finite m we obtain the bound \(d\le 2^{m/2}\) when T is finite, and \(d\le 2^{(m-2)/2}+2\) when T is infinite. For countably infinite m the bound is \(d\le 2^m.\) This relates to a question of Babai, who asked whether there existed a function f(d) such that every connected, locally finite graph G with maximum valence d has a 2-coloring of its vertices that is only preserved by the identity automorphism if the minimum number m of vertices moved by each non-identity automorphisms of G is at least \(m\ge f(d)\). Our results give a positive answer for trees. The trees need not be locally finite, their maximal valence can be \(2^{\aleph _0}\). For finite m we also extend our results to asymmetrizing trees by more than two colors.

设T为有限树或无穷大树，并且m为由T的非恒等同自同构移动的最小顶点数。我们给出的最大价界限d的牛逼在于保证一个顶点的着色存在ŧ与仅由身份构保留两种颜色。对于有限米我们得到结合\（d \文件2 ^ {M / 2} \）时Ť是有限的，和\（d \文件2 ^ {（M-2）/ 2} 2 \）时Ť是无限的。对于无穷大的m，边界为\（d \ le 2 ^ m。\）这与Babai的问题有关，Babai询问是否存在函数f（d），使得每个连接的具有最大价d的局部有限图G的顶点都具有2色，只有当G的每个不等式自同构移动的顶点的最小数量m至少为\（m \ ge f（d）\）。我们的结果为树木提供了肯定的答案。这些树不必是局部有限的，它们的最大价可以是\（2 ^ {\ aleph _0} \）。对于有限的m，我们还将结果扩展到使用不止两种颜色的不对称树。