# Monthly Archives: January 2016

## The rank and the kernel of a rayless tree

To prove Proposition 1.2 in the previous post on branches, we will furthermore work with the notion of rank of a rayless tree, i.e., a rank-function mapping any rayless tree to an ordinal number. For every rayless tree we define … Continue reading

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## Branches

To prove Proposition 1.2, we will work with the following notion. Definition 1.3. Suppose is a graph, a nonempty subgraph of , and a connected component of , the latter being the graph obtained by deleting every vertex of (and … Continue reading

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## Shaping by pruning: one shape-changing leaf implies infinitely-many shapes obtainable by finitely-often chopping off leaves.

We start with the shape of rayless trees. Our first goal is to prove the following statement. Proposition 1.2 (known since the 1980s). Suppose is an infinite rayless tree with one distinguished vertex, called its root. If there exists at … Continue reading

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## Definition 1.1 (tree, ray, rayless)

Here, by tree we mean a connected acyclic simple graph in the usual sense of contemporary graph theory. I.e., a tree is a pair with an arbitrary set, a subset of the set of -element subsets of , such that … Continue reading

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