Trees with large numbers of subtrees

ABSTRACT The number of subtrees in a graph is related to the reliability of a network with possible vertex failure and edge failure. It is known that networks with larger number of subtrees would be more reliable. Therefore, for given families of graphs, determining the graphs with maximum number of subtrees within the family becomes important. For a tree T of n vertices, let denote the number of all subtrees in T. Yan and Yeh gave a linear-time algorithm to count the number of subtrees in T and determined the tree of diameter d and order n with the maximum number of subtrees. In this paper, we present two recursive relations to count the number of subtrees of trees. We also characterize all trees of n vertices with .

[1]  Hua Wang,et al.  Binary trees with the largest number of subtrees , 2007, Discret. Appl. Math..

[2]  Borivoj Melichar,et al.  Computing all subtree repeats in ordered trees , 2012, Inf. Process. Lett..

[3]  Derek H. Smith.  Optimally Reliable Graphs for Both Vertex and Edge Failures , 1993, Comb. Probab. Comput..

[4]  Hua Wang,et al.  Extremal values of ratios: Distance problems vs. subtree problems in trees II , 2014, Discret. Math..

[5]  Scott Makeig,et al.  Enumeration of BC-subtrees of trees , 2006 .

[6]  Yeong-Nan Yeh,et al.  Enumeration of subtrees of trees , 2006, Theor. Comput. Sci..

[7]  Derek Smith,et al.  Optimally reliable graphs for both edge and vertex failures , 1986, Networks.

[8]  Hua Wang,et al.  Largest Number of Subtrees of Trees with a Given Maximum Degree , 2008, SIAM J. Discret. Math..

[9]  Maria Axenovich,et al.  Unavoidable subtrees , 2013, Discret. Math..

[10]  Atsuhiro Takasu,et al.  Author's Personal Copy Theoretical Computer Science Approximation and Parameterized Algorithms for Common Subtrees and Edit Distance between Unordered Trees , 2022 .

[11]  Daniel Gray,et al.  Trees with the Most Subtrees -- An Algorithmic Approach , 2012 .

[12]  Eberhard Triesch,et al.  Superdominance order and distance of trees with bounded maximum degree , 2003, Discret. Appl. Math..

[13]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[14]  Daniel M. Martin,et al.  The maximum agreement subtree problem , 2013, Discret. Appl. Math..

[15]  John Haslegrave,et al.  Extremal results on average subtree density of series-reduced trees , 2013, J. Comb. Theory, Ser. B.

[16]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[17]  Louis V. Quintas,et al.  A characterization of graphs of diameter two , 1988 .

[18]  László A. Székely,et al.  On subtrees of trees , 2005, Adv. Appl. Math..

[19]  Hua Wang,et al.  The Number of Subtrees of Trees with Given Degree Sequence , 2012, J. Graph Theory.

[20]  S. J. Cyvin,et al.  Kekule Structures in Benzenoid Hydrocarbons , 1988 .