We study protected nodes in $m$-ary search trees, by putting them in context of generalised Pólya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m $-ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $ m $; we conjecture that the method yields an asymptotically normal distribution for all $mleq 26$. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler Pólya urn (that is similar to the one that has earlier been used to study the total number of nodes in $ m $-ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $ mleq 26 $.
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机译：通过将广义Pólyaurns的背景下，我们将受保护的节点学习为M $ m $ m $ m $ m $ m $ m $ m $ m $。我们表明，在随机三元搜索树中，两种保护节点（既不叶子的节点也不是叶子的父母）是渐近正常的。这些方法原则上适用于$ M $ M $的搜索树，也是较大的$ M $，尽管计算中使用的矩阵的大小以$ M $迅速增长;我们猜测该方法为所有$ M LEQ 26美元产生渐近正常分布。一致的节点及其补充，即叶子，更容易分析。通过使用更简单的PólyaURN（类似于早期用于研究$ M $ M $ Meet搜索树中的节点总数），我们证明了一个受保护节点的数量的正常限制法律和所有$ m LEQ 26 $的叶子数。
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