A trivalent tree is an undirected tree where the degree of every internal vertex is three. We consider "topological subtrees'', which are the trivalent trees induced from a subset of k out of the n leaves, by taking the minimal subtree that spans them and suppressing any vertex of degree two. These subtrees appear in several applications and algorithmic problems, such as in Phylogenetics. Specifically, we consider the "subtree density'' which is the normalized number of copies of a given subtree with k leaves in a trivalent tree with n leaves.

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The talk will discuss extremal questions, such as what is the maximum possible density of a fixed subtree asymptotically as n tends to infinity, as well as probabilistic questions on subtree densities in large random trivalent trees. We introduce a new framework to discuss the above problems, a limit object for trivalent trees, called treeon.

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Advisor: Chaim Even-Zohar