Speaker : Prof. Shu-Chung Liu (National Hsinchu University of Education) Not just even and odd on plane trees 2014-07-10 (Thu) 14:00 - 16:00 Lecture Hall, Inst. of Mathematics 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. The conjecture was proved by Eu, Liu and Yeh in 2003 using three different methods: generating functions, induction and a two-to-one mapping. In this work, we apply the two-to-one mapping in the paper of Eu, Liu and Yeh to explore more properties of the rooted plane trees. First of all, we find that it is not just about odd over all rooted plane trees with n edges. Let $k\ge 1$. Actually, the number of vertices with degree $k$ is twice the number of vertices with out-degree $k$. 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 odd over all rooted plane trees with $n$ edges equals the number of the non-first children. Some other countings will be demonstrated in this talk.