Discrete Mathematics by K. Vesztergombi & J. Pelikán & L. Lovász

Author:K. Vesztergombi & J. Pelikán & L. Lovász
Language: eng
Format: epub
Publisher: Springer-Verlag Wien 2012
Published: 2014-06-03T16:00:00+00:00

The Tree-growing Procedure can be used to establish a number of properties of trees. Perhaps most important of these concerns the number of edges. How many edges does a tree have? Of course, this depends on the number of nodes; but surprisingly, it depends only on the number of nodes:

Theorem 8.2.3 Every tree on n nodes has n − 1 edges.

Proof. Indeed, we start with one more node (1) than edge (0), and at each step, one new node and one new edge are added, so this difference of 1 is maintained. □

8.2.2 Let G be a tree, which we consider as the network of roads in a medieval country, with castles as nodes. The king lives at node r. On a certain day, the lord of each castle sets out to visit the king. Argue carefully that soon after they have left their castles, there will be exactly one lord on each edge. Give a proof of Theorem 8.2.3 based on this.

8.2.3 If we delete a node v from a tree (together with all edges that end there), we get a graph whose connected components are trees. We call these connected components the branches at node v. Prove that every tree has a node such that every branch at this node contains at most half the nodes of the tree.

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