关于Prüfer码的局部性
On the locality of the Prüfer code
摘要：
Prüfer码是顶点集$[n]$上的树和长度为$n-2$的集合$[n]$上的字符串（$n$阶Prüfer字符串）之间的双射。在本文中，我们研究的“局部性”的性质的Prüfer码，即改变的Prüfer字符串的元素的结构上的相应的树的效果。我们对两棵树之间的距离$T,T^*$的度量是$Δ(T,T^*)=n-1-| E(T)\cap E(T^*)|$。我们随机地改变树$T$的Prüfer字符串的第个元素$μ$，将其改变为树$T^*$，并且我们渐近地估计这导致$\ell$边改变的概率，即$P(Δ=\ell | μ).$。我们发现P（Δ= \ell| μ）$ is on the order of $ n^{-1/3+o（1）} $ for any integer $ \ell >1，$ and that $ P（Δ=1| μ）=（1-μ/n）^2+o（1）。$ This result implies that the probability of a `perfect' mutation in the Prüfer code (one for which $ Δ（T，T^*）=1 $) is $ 1/3.$
英文摘要：
The Prüfer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Prüfer strings of order $n$). In this paper we examine the `locality' properties of the Prüfer code, i.e. the effect of changing an element of the Prüfer string on the structure of the corresponding tree. Our measure for the distance between two trees $T,T^*$ is $Δ(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $μ$th element of the Prüfer string of the tree $T$, changing it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\ell$ edges, i.e. $P(Δ=\ell | μ).$ We find that P(Δ=\ell | μ)$ is on the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(Δ=1 | μ)=(1-μ/n)^2+o(1).$ This result implies that the probability of a `perfect' mutation in the Prüfer code (one for which $Δ(T,T^*)=1$) is $1/3.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
