On the mean subtree order of trees under edge contraction

For a treeT $T$ , the mean subtree order ofT $T$ is the average order of a subtree ofT $T$ . In 1984, Jamison conjectured that the mean subtree order ofT $T$ decreases by at least 1/3 after contracting an edge inT $T$ . In this article we prove this conjecture in the special case that the contracted edge is a pendant edge. From this result, we have a new proof of the established fact that the pathP n${P}_{n}$ has the minimum mean subtree order among all trees of ordern $n$ . Moreover, a sharp lower bound is derived for the difference between the mean subtree orders of a treeT $T$ and a proper subtreeS $S$ (ofT $T$ ), which is also used to determine the tree with second‐smallest mean subtree order among all trees of ordern $n$ .