Abstract:
 Deutsch and Shapiro gave a conjecture in 2001 that the number of
vertices with odd degree is twice the number of vertices with odd
outdegree 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 twotoone
mapping. In this work, we apply the twotoone 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
outdegree $k$. 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 nonfirst children. Some
other countings will be demonstrated in this talk.
