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Let $mathcal{O}_n$ be the set of ordered labeled trees on ${0,...,n}$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study a new refinement $mathcal{O}_{n,k}$ of $mathcal{O}_n$, which is the set of ordered labeled trees whose maximal decreasing subtree has $k+1$ vertices.
Let $mathcal{T}^{(p)}_n$ be the set of $p$-ary labeled trees on ${1,2,dots,n}$. A maximal decreasing subtree of an $p$-ary labeled tree is defined by the maximal $p$-ary subtree from the root with all edges being decreasing. In this paper, we study a
For a labeled tree on the vertex set $set{1,2,ldots,n}$, the local direction of each edge $(i,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a ve
We use a sign-reversing involution to show that trees on the vertex set [n], considered to be rooted at 1, in which no vertex has exactly one child are counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1). This result corrects
In this paper, we present an involution on some kind of colored $k$-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers $C_{k,gamma}(n)=frac{gamma}{k n+gamma}{k n+gammachoose n}$. From the c
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $ge k$ at level $ge ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.