(This post is part of the Understanding GHOSTDAG series)

1. Draw all rooted trees with exactly five vertices.

2. Show that a rooted tree with $n$ vertices must have exactly $n-1$ arrows.

3. * A directed graph is (weakly) connected if for any two vertices it is possible to reach from one to the other by walking along arrows, but allowing ourselves to go against the direction. Show that any weakly connected directed graph with $n$ vertices and $n-1$ arrows is a tree. Show that a weakly connected rooted DAG is a tree if and only if removing any arrow will disconnect the graph.

4. A rooted tree is called binary if there are at most two arrows going into an vertex. The height of a tip is the number of arrows crossed when traversing from the tip to the root. The height of the tree is the height of the highest tip. What is the maximum number of tips a binary tree of height $h$ could have? What is the minimal?

5. A rooted tree with 14 vertices has the property that all vertices but the tips and the root have four or five arrows going into them, how many vertices have four arrows going into them?

6. Draw a block tree that contains a contradiction with the following property: it is impossible to HCR switch the selected chain from one side of the conflict to the other without adding at least ten blocks, but it is possible to do it for the GHOST rule by only adding a single block (you get to decide which side wins in case of a tie). Draw a tree with the same property when the roles of HCR and GHOST are reversed.

7. * Complete the analysis of the attack on the longest chain rule.