In this paper, we show that for any computable ordinal α, there exists a computable tree of rank $\alpha +1$ with strong degree of categoricity ${\mathbf{0}}^{(2\alpha )}$ if α is finite, and with strong degree of categoricity ${\mathbf{0}}^{(2\alpha +1)}$ if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with strong degree of categoricity ${\mathbf{0}}^{(\alpha )}$ (which equals ${\mathbf{0}}^{(2\alpha )}$). It follows from our proofs that, for every computable ordinal $\alpha >0$, the isomorphism problem for trees of rank α is ${\mathrm{\Pi }}_{2\alpha }^0$‐complete.

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树的分类度和同构问题

在本文中，我们表明对于任何可计算的序数α，都存在可计算的等级树 $\alpha +\mathrm{1个}$ 具有高度的分类性 ${\mathbf{0}}^{（2\alpha ）}$ 如果α是有限的并且具有强烈的分类性 ${\mathbf{0}}^{（2\alpha +\mathrm{1个}）}$如果α是无限的。实际上，这些是此类树的最大可能分类度。对于可计算的极限序数α，我们表明存在一类具有高度分类性的可计算树${\mathbf{0}}^{（\alpha ）}$ （等于 ${\mathbf{0}}^{（2\alpha ）}$）。根据我们的证明，对于每个可计算序数$\alpha >0$，等级为α的树的同构问题是 ${\mathrm{\Pi }}_{2\alpha }^0$-完成。