The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history of growing trees, based on exact results on root finding. We show that our linear-time algorithm produces the exact stepwise most probable history in a broad class of tree growth models. Our formulation is able to treat very large trees and therefore allows us to make reliable numerical observations regarding the possibility of root inference and history reconstruction in growing trees. We obtain the general formula $\langle ln\mathcal{N}\rangle \cong NlnN-cN$ for the size dependence of the mean logarithmic number of possible histories of a given tree, a quantity that largely determines the reconstructability of tree histories. We also reveal an uncertainty principle: a relationship between the inferability of the root and that of the complete history, indicating that there is a tradeoff between the two tasks; the root and the complete history cannot both be inferred with high accuracy at the same time.

中文翻译：

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在树的替代历史中进行选择

不断发展的网络结构包含有关其过去的信息。但是，有效地提取此信息通常是一个困难的挑战。我们根据根系寻找的确切结果，制定了一种快速有效的方法来估计最可能的树木生长史。我们证明了我们的线性时间算法在广泛的树木生长模型中产生了精确的逐步最可能的历史。我们的配方能够处理非常大的树木，因此使我们能够对正在生长的树木进行根推断和历史重建的可能性进行可靠的数值观察。我们得到了一般公式$〈ln\mathcal{ñ}〉\cong ñlnñ-Cñ$对于给定树的可能历史的平均对数数量的大小依赖性，该数量在很大程度上决定了树历史的可重构性。我们还揭示了不确定性原则：根的可推断性与完整历史的可推断性之间的关系，表明这两个任务之间存在权衡；不能同时高精度地推断出根和完整历史。