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Let $T_{n}$ be the set of rooted labeled trees on $set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $T_{n,k}$ of $T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.
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.
We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other fields. In
The Perron value $rho(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the
The greedy tree $mathcal{G}(D)$ and the $mathcal{M}$-tree $mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of inva
We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees, we show th