藉由Deutsch和Shapiro在2001年提出『在所有n邊的有根平面樹中，擁有odd degree之點數為odd out-degree點數的兩倍』這個猜測，而2003年游，劉和葉三位學者對這個性質提出了三個觀點的證明中。我們藉由擴充該定理性質且使用二對一個組合對應的證明方法，使得該性質擴充於更多情境下。在本篇文章中，我們發現當 k≥1， 在所有n邊的有根平面樹中，degree等於k之點數為out-degree等於k點數的兩倍；以及在所有n邊的平面樹中，out-degree大於0 的點數與 out-degree等於 0 （葉子點）的點數是一樣多的。我們發現在該定理下，有根平面樹不僅僅只有奇或偶的性質，還有存在著許多其他的性質。

Deutsch and Shapiro gave a conjecture in 2001 that the number of vertices with odd degree is twice the number of vertices with odd out-degree over all rooted plane trees with n edges. This conjecture was proved by Eu-Liu-Yeh in 2003 by three different ways: generating functions, induction and a two-to-one mapping. In this thesis we apply the two-to-one mapping and explore more properties of the rooted plane trees. First of all, we find that it is not just about the oddness over all rooted plane trees with n edges. Indeed, for k≥1, the number of vertices with degree k is twice the number of vertices with out-degree k. Second, the alternating sum of the numbers of vertices according to different ranks is 0. The third main result is that the number of all first children over all rooted plane trees with n edges equals the number of the non-first children.

[1] R. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge (1999).
[2] R. Stanley, Catalan Addendum, http://www-math.mit.edu/~rstan/ec/catadd.pdf
[3] E. Catalan, Note sur une equation aux differences, J. Math. Pures Appl. 3-1 (1838) 508-516.
[4] J.-J. Luo. "Ming Antu and His Power Series Expansions". Institute for the History of Science, Inner Mongolia Normal University; Institute of Science, Technology and Culture, Zhejiang University. Retrieved 26 (2012).
[5] 羅見今, 明安圖和他的冪級數展開式, 數學傳播第34卷第 1期 (2010) 65-73.
[6] E. Deutsch, L. Shapiro, A survey of the 3ne numbers, Discrete Math. 241 (2001) 241–265.
[7] S.-P. Eu, S.-C. Liu, and Y.-N. Yeh. Odd or even on plane trees. Discrete Math., 281 (2004) 189–196.
[8] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Sign-balance identities of Adin–Roichman type on 321-avoiding alternating permutations. Discrete Math., 312 (2012) 2228–2237.