A tale of trees and leaves : subtrees and local convergence

Sammanfattning: Given a tree T, a subtree in T is a subgraph that is a tree itself. The set of subtrees in a tree is related to many important graph parameters, one of which is the mean subtree order. Jamison initiated and laid the groundwork of the study of mean subtree order in the early 1980s and raised in total seven conjectures and open questions, on which the first two projects in the licentiate are based. In the first project, we will give a descrip- tion of the subtrees that achieve maximal/minimal local mean among all the subtrees of the same order. In the second project, we provide a proof that the mean subtree order decreases at least 1/3 in contracting an edge of a tree, which closed one of two conjectures that had remained open. Lastly, we study the asymptotic behavior of the number of subtrees for a sequence of trees that converges in the Benjamini–Schramm sense. 

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