Trees

Abstract

A tree is a connected graph that contains no cycle. In this chapter we know some properties of trees which are useful for solving computational problems on trees.

Exercises

• 1. Show that a forest with n vertices and k components has $n-k$ edges.

• 2. Show that a forest with n vertices and m edges contains $n-m$ components.

• 3. Show that a graph with n vertices, m edges, and $n-m$ components is acyclic.

• 4. Prove the correctness of Kruskal’s algorithm described in Section  4.4.

• 5. Draw all labeled trees with four vertices.

• 6. Construct all labeled spanning trees of $K_4$. How many of them are non-isomorphic?

• 7. Compute the ecentricities of the vertices in the graph in Fig. 4.14.

• 8. Find the centers of the trees in Figs. 4.15(a) and (b).

• 9. Show that the center of a tree is a single vertex if the diameter of the tree is even.

• 10. Compute the Prüffer’s code for Figs. 4.15(a) and (b).

• 11. Construct the tree corresponding to Prüffer’s code 1,2,2,7,6,6,5.

• 12. Find a graceful labeling of the tree in Fig. 4.15(b).

• 13. Develop an algorithm for computing a graceful labeling of a caterpillar.

• 14. Find a graceful labeling of the tree in Fig. 4.16. Can you develop an algorithm for finding graceful labelings of this type of trees?

• 15. Show that every tree containing a vertex of degree k contains at least k leaves.