Trees
10.1 Prove part (b) of theorem 10.1.
10.2 Prove that connecting two nodes and in a graph by a new edge creates a new cycle if and only if and are in the same connected component of .
10.3 Prove that in a tree, every two nodes can be connected by a unique path. Conversely, prove that if a graph has the property that every two nodes can be connected by a path, and there is only one connecting path for each pair, then the graph is a tree.
10.4 Apply the argument above to find a second node of degree 1.
10.5 Let be a tree, which we consider as the network of roads in a medieval country, with castles as nodes. The King lives at node . On a certain day, the lord of each castle sets out to visit the King. Argue carefully that soon after they leave their castles, there will be exactly one lord on each edge. Give a proof of Theorem 10.4 based on this.
10.6 If we delete a node 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 . Prove that every tree has a node such that every branch at this node contains at most half the nodes of the tree.
10.7 Find all unlabelled trees on 2, 3, 4 and 5 nodes. How many labelled trees do you get from each? Use this to find the number of labelled trees on 2, 3, 4 and 5 nodes.
10.8 How many labelled trees on nodes are stars? How many are paths?
10.9 Consider the following “codes”: (0, 1, 2, 3, 4, 5, 6, 7); (7, 6, 5, 4, 3, 2, 1, 0); (0, 0, 0, 0, 0, 0, 0, 0); (2, 3, 1, 2, 3, 1, 2, 3). Which of these are “father codes” of trees?
10.10 Prove, based on the “father code” method of storing trees, that the number of labelled trees on nodes is at most .
10.11 Complete the proof.
10.12 Does there exist an unlabelled tree with planar code
(a) 1111111100000000;
(b) 1010101010101010;
(c) 1100011100?
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