隨機樹節點的外出數度

Abstract

：在這個計畫中，我們想要研究隨機recursive樹裡外出數度為k的節點數目。 從一些關於unrooted unlabelled隨機樹及給定大小為k且位於隨機樹邊緣的子樹數 的最近研究結果顯示，我們猜測上述數目的期限分佈會隨者k的成長，從常態分 佈變換成Poisson分佈。我們提出一個方法，該方法不僅很可能可以證實此猜測， 也很可能可以更仔細地解釋此相變現象。除了隨機recursive樹之外，我們也想要 研究是否其他的隨機樹產生類似的分佈相變現象。

In this project, we will investigate the number of nodes with out-degree k in random recursive trees of size n. Recent results on unrooted unlabelled trees and on the number of subtrees at the fringe of random trees suggest that the limit law of the above quantity should undergo a phase change from normal to Poisson as k varies.We propose a method which should enable us to prove this conjecture as well as give a more detailed description of the phase change. Moreover, we plan to demonstrate that this phenomena exhibits some generality, i.e., a similar phase change is expected to hold for many other classes of random trees.