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We find a simple, closed formula for the proportion of vertices which are k-protected in all unlabeled rooted plane trees on n vertices. We also find that, as n goes to infinity, the average rank of a random vertex in a tree of size n approaches 0.727649, and the average rank of the root of a tree of size n approaches 1.62297.
The Erdős–Sós Conjecture states that every graph with average degree more than $$k-2$$ k - 2 contains all trees of order k as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an (n, m)-bipartite graph which does not contain all (k, l)-bipartite trees for given integers $$n\ge m$$ n ≥ m and $$k\ge l$$ k ≥ l . In...
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