On the Local and Global Means of Subtree Orders |
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| Authors: | Stephan Wagner Hua Wang |
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| Affiliation: | 1. DEPARTMENT OF MATHEMATICAL SCIENCES, STELLENBOSCH UNIVERSITY, MATIELAND, SOUTH AFRICAContract grant sponsors: National Research Foundation of South Africa;2. Contract grant number: 70560 (to S. W.);3. Contract grant sponsors: Simons Foundation;4. Contract grant number: 245307 (to H. W.).;5. DEPARTMENT OF MATHEMATICAL SCIENCES, GEORGIA SOUTHERN UNIVERSITY, STATESBORO, GA |
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| Abstract: | The global mean of subtrees of a tree is the average order (i.e., average number of vertices) of its subtrees. Analogously, the local mean of a vertex in a tree is the average order of subtrees containing this vertex. In the comprehensive study of these concepts by Jamison (J Combin Theory Ser B 35 (1983), 207–223 and J Combin Theory Ser B 37 (1984), 70–78), several open questions were proposed. One of them asks if the largest local mean always occurs at a leaf vertex. Another asks if it is true that the local mean of any vertex of any tree is at most twice the global mean. In this note, we answer the first question by showing that the largest local mean always occurs at a leaf or a vertex of degree 2 and that both cases are possible. With this result, a positive answer to the second question is provided. We also show some related results on local mean and global mean of trees. |
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| Keywords: | subtrees mean order local mean global mean homeomorphically irreducible trees |
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