条件Galton-Watson树的Horton-Strahler数
The Horton-Strahler Number of Conditioned Galton-Watson Trees
AI总结：
本文研究条件Galton-Watson树的Horton-Strahler数，证明其在临界有限方差条件下以概率按$\frac{1}{2}\log_2 n$增长，并推广到刚性Horton-Strahler数和$k$元寄存器函数，得到类似渐近结果。
摘要：
Horton-Strahler数是衡量树的分支复杂性的度量，在文献中也被称为寄存器函数。我们证明，对于具有有限方差且被条件化为大小为$n$的临界Galton-Watson树，其Horton-Strahler数以概率按$\frac{1}{2}\log_2 n$增长。我们进一步定义了该数的若干推广，其中包括刚性Horton-Strahler数和$k$元寄存器函数，并针对这些推广证明了与标准情形类似的渐近结果。
英文摘要：
The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case.
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。
